![Journal of Functional Analysis 258 (2010) 1–19
www.elsevier.com/locate/jfa
On OL∞ structure of nuclear, quasidiagonal
C∗-algebras
Caleb Eckhardt
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, United States
Received 25 June 2008; accepted 2 October 2009
Communicated by Alain Connes
Abstract
We continue the study of OL∞ structure of nuclear C∗-algebras initiated by Junge, Ozawa and Ruan.
In particular, we prove if OL∞(A) < 1.005, then A has a separating family of irreducible, stably fi-
nite representations. As an application we give examples of nuclear, quasidiagonal C∗-algebras A with
OL∞(A) > 1.
© 2009 Elsevier Inc. All rights reserved.
Keywords: Operator spaces; C∗-algebras; Quasidiagonal; Nuclear C∗-algebras
1. Introduction
This paper continues the study of OL∞-structure of nuclear C∗-algebras initiated by Junge,
Ozawa and Ruan in [8]. Before describing the contents of this paper, we recall the necessary
definitions and results.
Let V and W be n-dimensional operator spaces and consider the completely bounded version
of Banach–Mazur distance:
dcb(V,W) = inf
ϕcb
ϕ−1
cb
: ϕ : V → W is a linear isomorphism
.
Let A be a C∗-algebra. For λ 1 we say that OL∞(A) ⩽ λ if for every finite-dimensional
subspace E ⊂ A, there exist a finite-dimensional C∗-algebra B and a subspace E ⊂ F ⊂ A such
E-mail address: ceckhard@uiuc.edu.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.jfa.2009.10.004](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-5-320.jpg)
![2 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
that dcb(F,B) ⩽ λ. Then define
OL∞(A) = inf
λ: OL∞(A) ⩽ λ
.
A is a rigid OL∞ space if for every 0 and every x1,...,xn ∈ A there is a finite-dimensional
C∗-algebra B and a complete isometry ϕ : B → A such that dist(xi,ϕ(B)) for i = 1,...,n.
OL∞ is an interesting invariant for C∗-algebras, particularly when one considers the interplay
between OL∞ and various approximation properties of C∗-algebras.
It follows easily from the definition, that if OL∞(A) ∞, then there is a net of matrix
algebras (Mni ) and linear maps αi : A → Mni , βi : Mni → A such that βiαi tends to the identity
on A pointwise and supi αicbβicb ∞. Pisier showed [11, Theorem 2.9] that this implies A
is nuclear. Conversely, it was shown in [8] if A is nuclear, then OL∞(A) ⩽ 6. This estimate was
improved in [7] when the authors showed that all nuclear C∗-algebras A have OL∞(A) ⩽ 3. So,
OL∞ is most useful when restricted to nuclear C∗-algebras.
Another important approximation property is quasidiagonality (QD). We refer the reader to
the survey article [5] for information on QD C∗-algebras. The following relationships between
QD and OL∞ were established in [8]:
A is a rigid OL∞ space
(i)
−
→ OL∞(A) = 1
(ii)
−
−
→ A is nuclear QD.
Blackadar and Kirchberg showed [3, Proposition 2.5] that all 3 of the above assertions are equiv-
alent if A is either simple or both prime and antiliminal. The main purpose of this paper is to
give examples showing that the converse of (ii) does not hold in general.
In Section 2 we prove the necessary technical results used throughout the paper. Section 3
contains our first counterexamples to (ii). Section 4 contains some results about permanence
properties about OL∞. In Section 5 we prove the main result that all unital C∗-algebras A with
OL∞(A) 1.005 have a separating family of irreducible, stably finite representations. This
provides a larger class of nuclear quasidiagonal C∗-algebras A with OL∞(A) 1, but also has
implications for the converse of (i) which we discuss at the end of the paper.
2. Technical lemmas
In this section, we gather some technical lemmas needed for Sections 3 and 4, and fix our
notation.
Throughout the paper, if H is a Hilbert space, we let B(H) denote the space of bounded
linear operators on H. For H n-dimensional we write 2(n), and Mn for B(2(n)). We write
ucp and cpc as shorthand for “unital completely positive” and “completely positive contraction”
respectively. For linear maps ϕ : V → W between operator spaces we write ϕ(n) for idMn ⊗
ϕ :Mn(V ) → Mn(W), and ϕcb = supnϕ(n). Furthermore if ϕ is injective, we write ϕ−1
for the norm of the map ϕ−1 : ϕ(V ) → V . We write ⊗ for the minimal tensor product of C∗-
algebras.
The following lemma is implicit in the proof of [8, Theorem 3.2].
Lemma 2.1. Let 0 δ 1/
√
2, and let A be a unital C∗-algebra with OL∞(A) 1 + δ2/2.
Let F ⊂ A be a finite subset. Then there is a finite-dimensional C∗-algebra B, a linear map
ϕ : B → A with ϕcb 1 + δ2/2 and a ucp map ψ : A → B such that F ⊂ ϕ(B) and
ψϕ − idBcb
1 + δ2
/2
2
δ2 + δ4/4
.](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-6-320.jpg)
![C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 3
In [8, Theorem 3.2] the authors require δ 1/16. The reason for this is to guarantee that ψ is
approximately multiplicative on F . We will not need approximate multiplicativity in this paper,
which is why we are able to relax this condition to δ 1/
√
2.
We need the following slight variation of Lemma 2.1.
Lemma 2.2. Let λ (1+
√
3
2 )1/2 and A be a unital C∗-algebra with OL∞(A) λ. Let F ⊂ A
be a finite subset. Then there is a finite-dimensional C∗-algebra B, a ucp map ψ : A → B and a
unital, self-adjoint map ϕ : B → A such that:
(i) ϕcb λ
1−λ
√
2(λ2−1)
.
(ii) F ⊂ ϕ(B).
(iii) ψϕ = idB.
(iv) ϕψ|F = idF .
Proof. Without loss of generality suppose F consists of positive elements. We apply Lemma 2.1
with λ = 1 + δ2/2 to obtain a finite-dimensional C∗-algebra B, a ucp map ψ : A → B, and a
linear map ϕ : B → A such that F ⊂ ϕ(B), ϕcb λ and ψϕ − idBcb λ
2(λ2 − 1) 1.
Then ψϕ is invertible in the Banach algebra of all completely bounded maps on B. Let ϕ =
ϕ(ψϕ)−1. Then
ϕ cb ⩽ ϕcb
(ψϕ)−1
cb
⩽ λ
1
1 − λ
2(λ2 − 1)
.
Then ϕ satisfies (i)–(iii). Moreover, since ψ is unital and ψϕ = idB, it follows that ϕ is unital.
Finally, let ϕ (x) = 1/2(ϕ (x) + ϕ (x∗)∗), for x ∈ B. Then ϕ is unital, self-adjoint and
ϕ cb ⩽ ϕ cb. Since ψ is positive, it follows that ψϕ = idB. To see (ii), let b ∈ B such
that ϕ (b) ∈ F . Since F consists of positive elements, b = ψϕ (b) ⩾ 0. Hence, ϕ (b) =
1/2(ϕ (b) + ϕ (b)∗) = ϕ (b) ∈ F . Condition (iv) is a consequence of (ii) and (iii). 2
Lemma 2.3. Let A be a unital C∗-algebra and let x ∈ A. Set
x1 =
x1 x
x∗ x1
∈ M2 ⊗ A. (2.1)
Then x1 = 2x.
Proof. Without loss of generality, assume that x = 1. Clearly x1 ⩽ 2. For the reverse in-
equality, suppose that A ⊂ B(H) unitally for some Hilbert space H. By spectral theory there is
a sequence of unit vectors (ηk) ⊂ H such that
lim
k→∞
x∗
xηk − ηk
= 0. (2.2)
For each k ∈ N set
ξk =
1
√
2
xηk
ηk
∈ H ⊕ H.](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-7-320.jpg)
![C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 5
Proof. By [13, Theorem 2.10] every bounded map ψ from an operator space into Mn is com-
pletely bounded with ψ(n) = ψcb. So, we may assume that k = n. Also by [13, Theo-
rem 2.10], to prove inequality (2.3) it suffices to show that for every x ∈ Mn ⊗ Mn of norm 1, we
have
ϕ
(n)
1 (x)
⩾ r −
1 − r2
1 − s
. (2.4)
By Wittstock’s extension theorem [9, Theorem 8.2], let
ψ : B(L1) ⊕ B(L2) → Mn
be an extension of ϕ−1 : ϕ(Mn) → Mn with
ψcb = ϕ−1cb = r−1. Let ψ = r
ψ. Then
ψcb = 1 and
ψϕ(x) = rx for all x ∈ Mn. (2.5)
By the factorization theorem for completely bounded maps [9, Theorem 8.4] there is a unital
representation (π,H) of
Mn ⊗
B(L1) ⊕ B(L2)
= B
L1 ⊗ 2
(n)
⊕ B
L2 ⊗ 2
(n)
and isometries S,T : 2(n) ⊗ 2(n) → H such that
T ∗
π(x)S = ψ(n)
(x) for every x ∈ B
L1 ⊗ 2
(n)
⊕ B
L2 ⊗ 2
(n)
. (2.6)
Let qL1 = π(1L1⊗2(n),0) ∈ B(H) and qL2 = π(0,1L2⊗2(n)) ∈ B(H).
We now show that the ranges of S and T are almost included in qL1 (H).
Let ξ1 ∈ 2(n) ⊗ 2(n) be a norm 1 eigenvector for a with eigenvalue 1. Let ω1 ∈ Mn ⊗ Mn
be the orthogonal projection onto Cξ1. Then ω1 ⩽ a. Since ϕ2 is cp, we have ϕ
(n)
2 (ω1) ⩽
ϕ
(n)
2 (a) = s. Extend ξ1 to an orthonormal basis ξ1,ξ2,...,ξn2 for 2(n) ⊗ 2(n). For i =
1,...,n2 define the rank 1 operators,
ωi(η) = η,ξi ξ1, for η ∈ 2
(n) ⊗ 2
(n).
Then
ωiω∗
j = δi,j ω1, for 1 ⩽ i,j ⩽ n2
. (2.7)
Let η =
n2
i=1 αiξi ∈ 2(n) ⊗ 2(n) of norm 1 and ωη =
n2
i=1 αiωi ∈ Mn ⊗ Mn. By (2.7) and
Lemma 2.5, it follows that
ϕ
(n)
2 (ωη)
=
ϕ
(n)
2 (ωη)ϕ
(n)
2 (ωη)∗
1/2
⩽
n2
i=1
|αi|2
ϕ
(n)
2 (ω1)
1/2
⩽ s1/2
. (2.8)](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-9-320.jpg)
![C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 7
We will be careful with our norm estimates throughout the paper. Thus, the technical nature
of Lemma 2.6. Colloquially it states; regardless of the value of n ∈ N, if ϕ is almost a complete
isometry, then either ϕ1 or ϕ2 is almost a complete isometry. In particular, we have:
Corollary 2.7. Let L1,L2,n and ϕ be as in Lemma 2.6, but with ϕ−1cb = r−1 125/124.
Then either ϕ1 or ϕ2 is injective, and
ϕ−1
i
cb
⩽
1 + (r − 1)1/3
−1
for either i = 1 or i = 2.
Proof. If ϕ2 is injective with ϕ−1
2 cb (1 + (r − 1)1/3)−1, we are done. If not, then there is
an x ∈ Mn ⊗ Mn of norm 1 such that ϕ
(n)
2 (x) 1 + (r − 1)1/3. Then Lemma 2.4 provides an
a ∈ Mn ⊗ Mn of norm 1 with a ⩾ 0 such that
ϕ
(n)
2 (a)
⩽
1
2
1 +
ϕ
(n)
2 (x)
⩽
1
2
2 + (r − 1)1/3
.
We now apply Lemma 2.6 with s = 1
2 (1 + ϕ
(n)
2 (x)) to obtain,
ϕ−1
1
cb
⩽ r −
1 − r2
1 − s
−1
⩽
1 + (r − 1)1/3
−1
,
which holds whenever 124/125 r ⩽ 1. 2
Finally, we recall 2 useful perturbation lemmas.
Lemma 2.8. (See [16, Proposition 1.19].) Let A be a unital C∗-algebra and N an injective von
Neumann algebra. Let ϕ : A → N be a unital self-adjoint map with ϕcb ⩽ 1 + for some
0. Then there is a ucp map t : A → N such that t − ϕcb ⩽ .
Lemma 2.9. (See [12, Lemma 2.13.2].) Let 0 1 and X be an operator space. Let (xi,
xi)n
i=1
be a biorthogonal system with xi ∈ X and
xi ∈ X∗. Let y1,...,yn ∈ X be such that
xixi − yi .
Then there is a complete isomorphism w : X → X such that w(yi) = xi and wcbw−1cb ⩽
1+
1− .
3. First examples
For 1 ⩽ λ (1+
√
3
2 )1/2, let
f (λ) =
λ
1 − λ
√
2(λ − 1)
, (3.1)](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-11-320.jpg)
![10 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
In [8] it was asked if there were any nuclear, quasidiagonal C∗-algebras A with OL∞(A) 1.
We give some examples of such algebras. Let λ satisfy (3.3).
Example 3.2. Let s ∈ B(2) denote the unilateral shift. Then, A = C∗(s ⊕ s∗) is nuclear
and quasidiagonal. Applying Theorem 3.1 with ρ : A → C∗(s), σ : A → C∗(s∗) and (xn) =
(s ⊕ s∗)n, we have OL∞(A) λ .
Before the author obtained Theorem 3.1, Narutaka Ozawa outlined for me an alternate
proof that OL∞(C∗(s ⊕ s∗)) 1. The proof was based on the observation that for any finite-
dimensional C∗-algebra B and any partial isometry v ∈ B, we have 1 − vv∗ Murray–von Neu-
mann equivalent to 1 − v∗v. But, if we let (eij ) denote matrix units for B(2) and T = s ⊕ s∗,
then T is a partial isometry and 1 − T ∗T = 0 ⊕ e11 and 1 − T T ∗ = e11 ⊕ 0. So, 1 − T ∗T and
1−T T ∗ are not Murray–von Neumann equivalent in C∗(s ⊕s∗) = B(2)⊕B(2). One can use
these facts and arguments similar to Lemmas 2.2 and 2.8 to show that OL∞(C∗(s ⊕ s∗)) 1.
Example 3.3. (See [6, Example IX.11.2].) Let D1 and D2 be commuting diagonal operators with
joint essential spectrum RP2
, the real projective plane. Let s be as in Example 3.2. Set
A = C∗
(s ⊕ D1,0 ⊕ D2).
Then, A is easily seen to be an extension of nuclear C∗-algebras and hence is nuclear. As is shown
in [6], A is quasidiagonal. Applying Theorem 3.1, with ρ : A → C∗(s), σ : A → C∗(D1,D2) and
(xn) = (s ⊕ N1)n, we have OL∞(A) λ .
4. Permanence properties
We now investigate a couple permanence properties of OL∞.
Let B ⊂ A be nuclear C∗-algebras with OL∞(A) = 1. In general, we do not have
OL∞(B) = 1. Indeed let B = C∗(s ⊕ s∗) from Example 3.2. It is easy to see that s ⊕ s∗ is
a compact perturbation of a unitary operator u ∈ B(2 ⊕ 2). Let A = C∗(u) + K(2 ⊕ 2).
Then A is nuclear and inner quasidiagonal [3, Definition 2.2]. By [3, Theorem 4.5], A is a
strong NF algebra, which is a rigid OL∞-space by [2, Theorem 6.1.1]. Hence OL∞(A) = 1, but
OL∞(B) 1.
In contrast to this situation, if B is an ideal we have the following:
Theorem 4.1. Let A be a unital C∗-algebra and J an ideal of A. If OL∞(A) = 1, then
OL∞(J) = 1.
Proof. Let 0 and E ⊂ J a finite-dimensional subspace. Without loss of generality suppose
E has a basis of positive elements x1,...,xn ∈ E with xi = 1 for each i = 1,...,n. Let
x1,...,
xn ∈ J∗ such that xi,
xj = δi,j . Set M =
xi.
Define
δ1(δ) = (1 − δ) − (1 −
√
δ )−1
1 − (1 − δ)2
for 0 ⩽ δ 1.
Note that δ1(δ) → 1 as δ → 0.
Choose δ 0 small enough so 2
√
δ ⩽ /M and (2δ1(δ) − 1)−1 ⩽ 1 + .](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-14-320.jpg)
![C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 11
By Lemma 2.2 we obtain a finite-dimensional C∗-algebra B =
N
i=1 Mni , a ucp map ψ :A →
B and a unital, self-adjoint map ϕ : B → A with ϕcb ⩽ 1 + δ such that ϕψ|E = idE and
ψϕ = idB.
We will construct a finite-dimensional subspace E ⊂
F ⊂ J∗∗ such that
F is almost com-
pletely isometric to a finite-dimensional C∗-algebra and then apply a key theorem from [8] to
obtain a subspace E ⊂ F ⊂ J such that F is almost completely isometric to a finite-dimensional
C∗-algebra.
Since A∗∗ is injective, Lemma 2.8 provides a ucp map t : B → A∗∗ such that t − ϕcb ⩽ δ.
Then t is injective and t−1cb ⩽ (1 − δ)−1.
Let p ∈ A∗∗ be the central projection such that pA∗∗ = J∗∗. Let t1 : B → J∗∗ be defined by
t1(x) = pt(x) and t2 : B → A∗∗ by t2(x) = (1 − p)t(x).
Returning to the C∗-algebra B, let q1,...,qN ∈ B be the minimal central projections such
that qiB ∼
= Mni . Let
I =
1 ⩽ i ⩽ N: sup
1⩽j⩽n
qiψ(xj )
⩽
√
δ
. (4.1)
Set q =
i /
∈I qi and C = qB.
We now show that t1 : C → J∗∗ is injective with t1|−1
C cb ⩽ (2δ1(δ) − 1)−1. We first show
t1 restricted to each summand of C is almost a complete isometry.
To this end, let Ic = {1,...,N}I and j ∈ Ic. Then there is an xi such that qj ψ(xi)
√
δ.
Since xi ∈ J ∩ E, we have
(1 − p)ϕψ(xi) = (1 − p)xi = 0. (4.2)
Since ψ is ucp,
qj ψ(xi)
−1
qj ψ(xi) ∈ qj B ∼
= Mnj (4.3)
is norm 1 and positive. Combining (4.2) and (4.3) we obtain
t2
qj ψ(xi)
−1
qj ψ(xi)
⩽
qj ψ(xi)
−1
t2
ψ(xi)
=
qj ψ(xi)
−1
(1 − p)tψ(xi)
⩽
qj ψ(xi)
−1
(1 − p)ϕψ(xi)
+ δ
⩽
√
δ.
We apply Lemma 2.6 to t1 : qj B → J∗∗ with
s =
t2
qj ψ(xi)
−1
qj ψ(xi)
⩽
√
δ and
r−1
=
t|−1
qj B
cb
⩽
t|−1
B
cb
⩽ (1 − δ)−1
to obtain
t1|−1
qj B
cb
⩽ δ1(δ)−1
for all j ∈ Ic
. (4.4)](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-15-320.jpg)
![12 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
Now, let k ∈ N be arbitrary and a =
j /
∈I(1k ⊗ qj )a ∈ Mk ⊗ C be positive. Since t1 is com-
pletely positive, by (4.4),
t
(k)
1 (a)
⩾ sup
j /
∈I
t
(k)
1
(1k ⊗ qj )a
⩾ δ1(δ)a.
By Lemma 2.4, t1 : C → J∗∗ is injective with t−1
1 cb ⩽ (2δ1(δ) − 1)−1.
t1(C) does not necessarily contain x1,...,xn. We fix this with a perturbation. Since pxi =
xi = ϕψ(xi) for i = 1,...,n, it follows from (4.1) that
xi − t1
ψ(xi)q
⩽
xi − t1ψ(xi)
+
√
δ
=
xi − ptψ(xi)
+
√
δ
⩽
xi − pϕψ(xi)
+ δ +
√
δ
= δ +
√
δ.
Set yi = t1(ψ(xi)q) ∈ J∗∗ for i = 1,...,n. Then,
n
i=1
xixi − yi ⩽ M(2
√
δ ) ⩽ .
By Lemma 2.9 there is a complete isomorphism w : J∗∗ → J∗∗ such that w(yi) = xi for
i = 1,...,n and wcbw−1cb ⩽ (1 + )/(1 − ).
Let
F = wt1(C) ⊂ J∗∗. Then E ⊂
F and
dcb(
F,C) ⩽
(1 + )
(1 − )
2δ1(δ) − 1
−1
(1 + )2
1 −
.
By [8, Theorem 4.3] there is a subspace F ⊂ J such that E ⊂ F and dcb(F,C)
(1 + )2(1 − )−1.
Since 0 was arbitrary, it follows that OL∞(J) = 1. 2
Remark 4.2. It is not known if Theorem 4.1 holds in general, i.e. if J is an ideal of A do we
always have OL∞(J) ⩽ OL∞(A)?
Remark 4.3. Blackadar and Kirchberg have shown [2, Proposition 6.1.7] that every hereditary
subalgebra of a rigid OL∞ space is also a rigid OL∞ space. It is not known if Theorem 4.1 can
be extended to include hereditary sub C∗-algebras.
Finally, we need the following Proposition for Section 5. For C∗-algebras A and B, let A B
denote the algebraic tensor product of A and B.
Proposition 4.4. Let A1 and A2 be nuclear C∗-algebras. Then
OL∞(A1 ⊗ A2) ⩽ OL∞(A1)OL∞(A2).](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-16-320.jpg)
![C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 13
Proof. Let E ⊂ A1 A2 be a finite-dimensional subspace and 0. For i = 1,2 choose finite-
dimensional subspaces Fi ⊂ Ai, finite-dimensional C∗-algebras Bi and linear isomorphisms
ϕi :Fi → Bi, such that
ϕicb
ϕ−1
i
cb
⩽ OL∞(Ai) + and E ⊂ F1 F2.
Let ⊗min denote the minimal operator space tensor product. Recall that for C∗-algebras the
minimal operator space tensor product coincides with the minimal C∗-tensor product (see [12,
p. 228]). Furthermore by [12, 2.1.3],
ϕ1 ⊗ ϕ2 : F1 ⊗min F2 → B1 ⊗ B2cb ⩽ ϕ1cbϕ2cb.
We have a similar inequality for (ϕ1 ⊗ ϕ2)−1 = ϕ−1
1 ⊗ ϕ−1
2 . Since A1 A2 is dense in A1 ⊗ A2,
it follows that
OL∞(A1 ⊗ A2) ⩽ inf
0
OL∞(A1) +
OL∞(A2) +
= OL∞(A1)OL∞(A2). 2
5. Irreducible representations and OL∞
This section contains the main theorem (Theorem 5.4). We first recall the necessary definitions
and prove some preliminary lemmas.
Definition 5.1. Let A be a unital C∗-algebra. Recall that x ∈ A is an isometry if x∗x = 1. An
isometry is called proper, if xx∗ = 1. A is called finite if it contains no proper isometries. A is
called stably finite if Mn ⊗ A is finite for every n ∈ N. We will call a representation π of A finite
(resp. stably finite) if A/ker(π) is finite (resp. stably finite).
Lemma 5.2. Let H be a separable Hilbert space and x ∈ B(H) be a proper isometry. Then there
is a unitary u ∈ B(H) such that (ux)n(ux)∗n → 0 strongly.
Proof. It is well known (see [6, Theorem V.2.1]) that there is a closed subspace K ⊂ H such
that relative to the decomposition H = K ⊕ K⊥, we have x = s ⊕ w where s ∈ B(K) is unitarily
equivalent to sα, the unilateral shift of order α (for some α = 1,2,...,∞), and w is a unitary
in B(K⊥). In particular sns∗n → 0 strongly in B(K).
Without loss of generality, assume that w = idK⊥ .
Suppose first that K⊥ is infinite-dimensional. Since x|K is a proper isometry, K is also
infinite-dimensional. Since H is separable, K ∼
= K⊥. Under this identification and relative to
the decomposition H = K ⊕ K, let
u =
0 1
1 0
∈ B(H).
Then for n ∈ N we have,
(ux)2n
(ux)∗2n
=
sns∗n 0
0 sns∗n](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-17-320.jpg)
![14 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
and
(ux)2n+1
(ux)∗(2n+1)
=
sns∗n 0
0 s(n+1)s∗(n+1) .
Hence, (ux)n(ux)∗n → 0 strongly.
Suppose now that dim(K⊥) = n ∞. Let {f1,...,fn} be an orthonormal basis for K⊥.
Since s is unitarily equivalent to a shift, let e1,...,en ∈ K be an orthonormal set such that
sei = ei+1 for i = 1,...,n − 1 and e1 ⊥ x(H).
Define u ∈ B(H) by u(ei) = fi and u(fi) = ei for i = 1,...,n and u(η) = η for η ⊥
span{e1,...,en,f1,...,fn}. Then u is unitary and
(ux)2n
(H) ⊥ span{e1,...,en,f1,...,fn}.
Hence for every k ⩾ 2n we have (ux)2n+k = xk(ux)2n. Therefore,
(ux)2n+k
(ux)∗(2n+k)
⩽
sk
⊕ 0K⊥
sk
⊕ 0K⊥
∗
→ 0 strongly. 2
We recall the following definitions (see [10, Section 4.1]).
Let A be a C∗-algebra. An ideal J of A is called primitive if J is the kernel of some (non-
zero) irreducible representation of A. Let Prim(A) denote the set of all primitive ideals of A. For
a subset X ⊂ Prim(A), and an ideal J of A let
ker(X) =
I∈X
I and hull(J) =
I ∈ Prim(A): J ⊂ I
.
Then Prim(A) is a topological space with closure operation X → hull(ker(X)) (see [10, Theo-
rem 4.1.3]).
The following is an easy consequence of [10, Theorem 4.1.3].
Lemma 5.3. Let A be a C∗-algebra and X ⊂ Prim(A). Then X is dense if and only if ker(X) =
{0}.
Theorem 5.4. Let A be a separable unital C∗-algebra with OL∞(A) λ , where λ satis-
fies (3.3). Then A has a separating family of irreducible, stably finite representations.
Proof. We first show that A has a separating family of irreducible, finite representations.
We assume that A does not have a separating family of irreducible, finite representations and
prove that OL∞(A) λ . Let
Y =
y ∈ A:
∃J ∈ Prim(A)
(y + J ∈ A/J is a proper isometry)
.
Then Y is not empty. For each y ∈ Y, let
O(y) =
J ∈ Prim(A):
1 − y∗
y
+ J
1/4 and
1 − yy∗
+ J
3/4
, (5.1)](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-18-320.jpg)
![C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 15
CO(y) = Prim(A) hull
ker
O(y)
.
We will now prove the following statement:
(∃y ∈ Y)
CO(y) is not dense in Prim(A)
. (5.2)
(If Prim(A) is Hausdorff, then (5.2) is immediate by [10, Proposition 4.4.5]. But Prim(A) is not
Hausdorff in general.)
Since A is separable, let (yn) ⊂ Y be a dense sequence.
Suppose that (5.2) does not hold. Then CO(yn) is a dense, open subset of Prim(A) for each
n ∈ N. Since Prim(A) is a Baire space, (see [10, Theorem 4.3.5]) the following set is dense in
Prim(A):
X =
∞
n=1
CO(yn).
If there is a J ∈ X such that A/J is not finite, then there is a y ∈ Y such that y + J is a proper
isometry. Then there is an n ∈ N such that
yny∗
n − yy∗
+
y∗
nyn − y∗
y
1/8.
But this implies that
J ∈ O(yn) ∩ X ⊂ hull
ker
O(yn)
∩ X = ∅.
Hence for every J ∈ X, A/J is finite. Since X is dense, ker(X) = {0} by Lemma 5.3. Then A
has a separating family of irreducible finite representations, a contradiction. This completes the
proof of (5.2).
We now build representations ρ and σ that satisfy Theorem 3.1. Let y ∈ Y satisfy (5.2).
For each J ∈ CO(y) let σJ be an irreducible representation of A such that ker(σJ ) = J . Let
σ =
J∈CO(y) σJ . Since CO(y) is not dense, we have
ker(σ) =
J∈CO(y)
J = ker
CO(y)
= {0}. (5.3)
Let {Ji}i∈I ⊂ O(y) be an at most countable subset such that
ker
{Ji}i∈I
= ker
O(y)
. (5.4)
For i ∈ I, let ρi be an irreducible representation of A such that ker(ρi) = Ji. Let ρ =
i∈I ρi.
By (5.3) and (5.4) we have
ker(ρ ⊕ σ) = ker
O(y)
∩
J∈CO(y)
J =
J∈Prim(A)
J = {0}. (5.5)](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-19-320.jpg)
![16 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
By definition (5.1), for every i ∈ I, we have 1 − ρi(y∗y) 1/4. Hence ρi(y) is left invertible
and ρi(y∗y) is invertible.
We note that ρi(y) is not right invertible. Indeed, if ρi(y) is right invertible, then there is a
unitary u ∈ ρi(A) such that ρi(y) = u|ρi(y)|. Then by (5.1) we have
3/4
1 − ρi
yy∗
=
1 − uρi
y∗
y
u∗
=
1 − ρi
y∗
y
1/4,
a contradiction.
For each i ∈ I, let
zi = ρi(y)
ρi
y∗
y
−1/2
.
Then z∗
i zi = 1, but ziz∗
i = 1 because ρi(y) is not right invertible. Hence, zi ∈ ρi(A) is a proper
isometry for each i ∈ I. Define the continuous function f : R+ → R+ by
f (t) =
8
3
√
3
t if 0 ⩽ t ⩽ 3/4,
t−1/2 if t 3/4.
Let
x = yf (y∗y) ∈ A. Since sp(ρi(y∗y)) ⊂ [3/4,1], it follows that ρi(
x ) = zi for each i ∈ I. Let
x ∈ A be norm 1 such that ρ(x) = ρ(
x ) (such a lifting is always possible, see [16, Remark 8.6]).
Let Hi denote the Hilbert space associated with ρi. For each i ∈ I, Lemma 5.2 provides a
unitary ui ∈ B(Hi) such that
(uizi)n
(uizi)∗n
→ 0 strongly in B(Hi), as n → ∞. (5.6)
Since each ρi has a different kernel, they are mutually inequivalent. So, by [10, Theorem 3.8.11]
ρ(A) =
i∈I
ρi(A) =
i∈I
B(Hi).
Set u =
i∈I ui. Since ρi(x) = zi, by (5.6) we have
uρ(x)
n
uρ(x)
∗n
→ 0 strongly in
i∈I
B(Hi).
By Kaplansky’s density theorem (see [14, Theorem II.4.11]) there is a sequence (uk) of unitaries
from ρ(A) such that uk → u in the strong* topology. From this we obtain sequences (kr) and
(nr) such that
ukr ρ(x)
nr
ukr ρ(x)
∗nr
→ 0 strongly as r → ∞. (5.7)
For each r ∈ N let xr ∈ A be norm 1 such that ρ(xr) = ukr . By (5.3) and (5.5) we apply Theo-
rem 3.1 with the sequence (xrx)∞
r=1 and deduce that OL∞(A) λ .
We now return to the general case. Suppose that OL∞(A) λ .
Let H be a separable, infinite-dimensional Hilbert space. Let K denote the compact oper-
ators on H and K1 be the unitization of K. Since K1 is an AF algebra, OL∞(K1) = 1. By
Proposition 4.4, OL∞(A ⊗ K1) ⩽ OL∞(A) λ .](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-20-320.jpg)
![C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 17
By the above proof there is a subset X ⊂ Prim(A ⊗ K1) with ker(X) = {0} and (A ⊗ K1)/J
finite for each J ∈ X.
By [1, IV.3.4.23],
Prim
A ⊗ K1
=
J ⊗ K1
+ A ⊗ I: J ∈ Prim(A), I = {0}, K
.
So, without loss of generality we may assume X = {Ji ⊗ K1}i∈I with Ji ∈ Prim(A). Since
K1 is exact, (A ⊗ K1)/(Ji ⊗ K1) = (A/Ji) ⊗ K1, so A/Ji is stably finite. Furthermore, by the
exactness of K1, we have
{0} =
i∈I
Ji ⊗ K1
=
i∈I
Ji ⊗ K1
.
So, ker({Ji}i∈I ) = {0}. 2
We are now in a position to give a new class of examples of nuclear, quasidiagonal C∗-
algebras A with OL∞(A) 1.
Example 5.5. Let A be a unital nuclear C∗-algebra without a separating family of irreducible
stably finite representations (in particular any non-finite nuclear, C∗-algebra). Let C(A)1 =
(C0(0,1] ⊗ A)1 be the unitization of the cone of A. Since A is nuclear, so is C(A)1. By [15,
Proposition 3] C(A)1 is quasidiagonal. For t ∈ (0,1], let It = {f ∈ C0(0,1]: f (t) = 0}. By
[1, IV.3.4.23] every non-essential primitive ideal of C(A)1 is of the form
It ⊗ A + C0(0,1] ⊗ J
for some J ∈ Prim(A) and 0 t ⩽ 1. Furthermore, by [1, IV.3.4.22],
C0(0,1] ⊗ A
/
It ⊗ A + C0(0,1] ⊗ J
∼
= A/J.
From this we deduce that C(A)1 cannot have a separating family of irreducible, stably finite
representations, hence OL∞(C(A)1) λ by Theorem 5.4.
6. Questions and remarks
Recall from the Introduction:
Question 6.1. (See [8, Question 6.1].) If OL∞(A) = 1, is A a rigid OL∞ space?
Blackadar and Kirchberg showed [3, Theorem 4.5] that a C∗-algebra A is nuclear and inner
quasidiagonal if and only if A is a strong NF algebra (see [2, Definition 5.2.1]). In [8] it was
shown that A is a strong NF algebra if and only if A is a rigid OL∞ space. Furthermore, by [3,
Proposition 2.4] any C∗-algebra with a separating family of irreducible quasidiagonal represen-
tations is inner quasidiagonal.
Therefore if there is a C∗-algebra A with OL∞(A) = 1, but which is not a rigid OL∞ space,
then A cannot have a separating family of irreducible, quasidiagonal representations, but A must](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-21-320.jpg)
![18 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
have a separating family of irreducible stably finite representations by Theorem 5.4. Let A ⊂
B(H) from Example 3.3. Then A + K(H) is stably finite and prime, hence has a faithful stably
finite representation. On the other hand by [4], the unique irreducible representation of A+K(H)
is not quasidiagonal. Hence, A + K(H) is a possible counterexample to Question 6.1.
Finally, recall the question raised by Blackadar and Kirchberg:
Question 6.2. (See [2, Question 7.4].) Is every nuclear stably finite C∗-algebra quasidiagonal?
There are some interesting relationships between Question 6.2 and OL∞ structure.
Proposition 6.3. Let A be either simple or both prime and antiliminal. If
1 OL∞(A)
1 +
√
5
2
1/2
then A is (nuclear) stably finite, but not quasidiagonal.
Proof. This follows from [3, Corollary 2.6] and [8, Theorem 3.4]. 2
In light of Theorem 5.4, we have the following similar relationship:
Proposition 6.4. Let A be a C∗-algebra such that every primitive quotient is antiliminal. If
1 OL∞(A) 1.005
then some quotient of A is (nuclear) stably finite, but not quasidiagonal.
Proof. This follows from [3, Corollary 2.6] and Theorem 5.4. 2
Acknowledgments
A portion of the work for this paper was completed while the author took part in the Thematic
Program on Operator Algebras at the Fields Institute in Toronto, ON in the Fall of 2007. I would
like to thank Narutaka Ozawa for a helpful discussion about this work and my advisor Zhong-Jin
Ruan for all his support.
References
[1] B. Blackadar, Operator algebras, in: Theory of C∗-Algebras and von Neumann Algebras, Operator Algebras and
Non-Commutative Geometry, III, in: Encyclopaedia Math. Sci., vol. 122, Springer-Verlag, Berlin, 2006.
[2] Bruce Blackadar, Eberhard Kirchberg, Generalized inductive limits of finite-dimensional C*-algebras, Math.
Ann. 307 (3) (1997) 343–380.
[3] Bruce Blackadar, Eberhard Kirchberg, Inner quasidiagonality and strong NF algebras, Pacific J. Math. 198 (2)
(2001) 307–329.
[4] L.G. Brown, The universal coefficient theorem for Ext and quasidiagonality, in: Operator Algebras and Group
Representations, vol. I, Neptun, 1980, in: Math. Stud. Monogr. Ser., vol. 17, Pitman, Boston, MA, 1984, pp. 60–64.
[5] Nathanial P. Brown, On quasidiagonal C∗-algebras, in: Operator Algebras and Applications, in: Adv. Stud. Pure
Math., vol. 38, Math. Soc. Japan, Tokyo, 2004, pp. 19–64.](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-22-320.jpg)
![C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 19
[6] Kenneth R. Davidson, C∗-Algebras by Example, Fields Inst. Monogr., vol. 6, American Mathematical Society,
Providence, RI, 1996.
[7] M. Junge, N.J. Nielsen, Zhong-Jin Ruan, Q. Xu, COLp spaces—the local structure of non-commutative Lp spaces,
Adv. Math. 187 (2) (2004) 257–319.
[8] Marius Junge, Narutaka Ozawa, Zhong-Jin Ruan, On OL∞ structures of nuclear C∗-algebras, Math. Ann. 325 (3)
(2003) 449–483.
[9] Vern Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Stud. Adv. Math., vol. 78, Cambridge
University Press, Cambridge, 2002.
[10] Gert K. Pedersen, C∗-Algebras and Their Automorphism Groups, London Math. Soc. Monogr. Ser., vol. 14, Aca-
demic Press Inc., London, 1979.
[11] Gilles Pisier, The operator Hilbert space OH, complex interpolation and tensor norms, Mem. Amer. Math.
Soc. 122 (585) (1996).
[12] Gilles Pisier, Introduction to Operator Space Theory, London Math. Soc. Lecture Note Ser., vol. 294, Cambridge
University Press, Cambridge, 2003.
[13] R.R. Smith, Completely bounded maps between C∗-algebras, J. London Math. Soc. (2) 27 (1) (1983) 157–166.
[14] M. Takesaki, Theory of Operator Algebras, I, Encyclopaedia Math. Sci., vol. 124, Springer-Verlag, Berlin, 2002.
Reprint of the first edition, 1979, Operator Algebras and Non-Commutative Geometry, vol. 5.
[15] Dan Voiculescu, A note on quasi-diagonal C∗-algebras and homotopy, Duke Math. J. 62 (2) (1991) 267–271.
[16] Simon Wassermann, Exact C∗-Algebras and Related Topics, Lect. Notes Ser., vol. 19, Seoul National University,
Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994.](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-23-320.jpg)
![Journal of Functional Analysis 258 (2010) 20–49
www.elsevier.com/locate/jfa
Groupoid normalizers of tensor products
Junsheng Fang a
, Roger R. Smith a,∗,1
, Stuart A. White b
,
Alan D. Wiggins c
a Department of Mathematics, Texas AM University, College Station, TX 77843, USA
b Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK
c Department of Mathematics and Statistics, University of Michigan–Dearborn, Dearborn, MI 48128, USA
Received 15 September 2008; accepted 7 October 2009
Communicated by N. Kalton
Abstract
We consider an inclusion B ⊆ M of finite von Neumann algebras satisfying B ∩ M ⊆ B. A partial
isometry v ∈ M is called a groupoid normalizer if vBv∗,v∗Bv ⊆ B. Given two such inclusions Bi ⊆ Mi,
i = 1,2, we find approximations to the groupoid normalizers of B1 ⊗ B2 in M1 ⊗ M2, from which we
deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal
to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are
given to show that this can fail without the hypothesis B
i ∩Mi ⊆ Bi, i = 1,2. We also prove a parallel result
where the groupoid normalizers are replaced by the intertwiners, those partial isometries v ∈ M satisfying
vBv∗ ⊆ B and v∗v,vv∗ ∈ B.
© 2009 Elsevier Inc. All rights reserved.
Keywords: Groupoid normalizer; Tensor product; von Neumann algebra; Finite factor
1. Introduction
The focus of this paper is an inclusion B ⊆ M of finite von Neumann algebras. Such inclusions
have a rich diverse history, first being studied by Dixmier [3] in the context of maximal abelian
subalgebras (masas) of II1 factors. These inclusions provided the basic building blocks for the
* Corresponding author.
E-mail addresses: jfang@math.tamu.edu (J. Fang), rsmith@math.tamu.edu (R.R. Smith), s.white@maths.gla.ac.uk
(S.A. White), adwiggin@umd.umich.edu (A.D. Wiggins).
1 Partially supported by a grant from the National Science Foundation.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.jfa.2009.10.005](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-24-320.jpg)
![J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 21
theory of subfactors developed by Jones in [9] and today they are a key component in the study of
structural properties of II1 factors using the deformation-rigidity techniques introduced by Popa
in [14].
In [3], Dixmier introduced a classification of masas in II1 factors using normalizers, defining
NM(B) = {u a unitary in M: uBu∗ = B}. A masa B ⊂ M is Cartan or regular if these normal-
izers generate M and singular if NM(B) ⊂ B. Feldman and Moore demonstrated the importance
of Cartan masas, and hence normalizers, in the study of II1 factors, showing that inclusions of
Cartan masas arise from measurable equivalence relations and that, up to orbit equivalence, these
relations determine the resulting inclusion [7,8].
Given two inclusions Bi ⊂ Mi of masas in II1 factors, it is immediate that an elementary
tensor u1 ⊗u2 of unitaries ui ∈ Mi normalizes the tensor product inclusion B = B1 ⊗B2 ⊂ M =
M1 ⊗ M2 if and only if each ui normalizes Bi. As a simple consequence, the tensor product
of Cartan masas is again Cartan. More generally, the operation of passing to the von Neumann
algebra generated by the normalizers was shown to commute with the tensor product operation
for masas inside II1 factors, in the sense that the equality
NM1 (B1)
⊗ NM2 (B2)
= NM1⊗M2
(B1 ⊗ B2)
(1.1)
holds. This was proved when both masas are singular in [19] and the general case was established
by Chifan in [1]. Since the containment from left to right in (1.1) is immediate, the problem in
both cases is to eliminate the possibility that some unexpected unitary in the tensor product
normalizes B1 ⊗ B2. This difficulty was overcome in [19] and [1] by employing techniques
of Popa [15] to analyse the basic construction algebra M,eB of Jones [9]. Beyond the masa
setting, (1.1) holds when each Bi satisfies B
i ∩ Mi = C1, the defining property of irreducible
subfactors. When each Bi has finite Jones index in Mi, the identity (1.1) can be deduced from
results of [13]. The infinite index case was established in [20], where every normalizing unitary
of such a tensor product of irreducible subfactors was shown to be of the form w(v1 ⊗v2), where
w is a unitary in B1 ⊗ B2 and each vi ∈ NMi (Bi). Some other situations where (1.1) holds are
discussed in [6].
For general inclusions Bi ⊆ Mi of finite von Neumann algebras, the commutation iden-
tity (1.1) can fail. Indeed, taking each Mi to be a copy of the 3 × 3 matrices and each
Bi
∼
= C ⊕ M2(C), one obtains inclusions with NMi (Bi) ⊂ Bi, yet there are non-trivial normaliz-
ers of B1 ⊗B2 inside M1 ⊗M2. This is due to the presence of partial isometries v in Mi Bi with
vBiv∗ ⊆ Bi and v∗Biv ⊆ Bi, as the non-trivial unitary normalizers of B1 ⊗ B2 can all be writ-
ten in the form
j xj (v1,j ⊗ v2,j ), where xj lie in B1 ⊗ B2 and the vi,j are partial isometries
with vi,j Biv∗
i,j ⊆ B and v∗
i,j Bivi,j ⊆ Bi. Defining the groupoid normalizers of a unital inclu-
sion B ⊂ M to be the set GNM(B) = {v a partial isometry in M: vBv∗ ⊆ B,v∗Bv ⊆ B}, the
example discussed above satisfies the commutation identity
GNM1 (B1)
⊗ GNM2 (B2)
= GNM1⊗M2
(B1 ⊗ B2)
. (1.2)
In this paper we examine groupoid normalizers of tensor product algebras, establishing (Corol-
lary 5.6) the identity (1.2) whenever Bi ⊆ Mi are inclusions of finite von Neumann algebras with
separable preduals satisfying B
i ∩ Mi ⊆ Bi for each i. In [4] Dye shows that every groupoid
normalizer v of a masa B in M is of the form v = ue for some projection e = v∗v ∈ B and
some unitary normalizer u of B in M, see also [18, Lemma 6.2.3]. The same result holds
by a direct computation when B is an irreducible subfactor of M, so that in these two cases](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-25-320.jpg)
![22 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
NM(B) = GNM(B) and (1.2) directly generalizes (1.1) established in [1] and [20] respec-
tively.
The following example shows why the hypothesis B
i ∩ Mi ⊆ Bi (which is satisfied by both
masas and irreducible subfactors) is necessary in this result.
Example 1.1. Consider the subalgebra
B =
α 0 0
0 α 0
0 0 β
: α,β ∈ C
⊆ M3,
and note that B ∩ M3 strictly contains B. A direct computation shows that GN(B) = M2 ⊕ C,
and so GNM3 (B) ⊗ GNM3 (B) ∼
= M4 ⊕ M2 ⊕ M2 ⊕ C. However, B ⊗ B is isomorphic to
CI4 ⊕ CI2 ⊕ CI2 ⊕ C inside M9, and GNM3⊗M3 (B ⊗ B) is M4 ⊕ M4 ⊕ C.
A new feature of [20] was the notion of one-sided normalizers of an irreducible inclusion
B ⊂ M of II1 factors, namely those unitaries u ∈ M with uBu∗ B. These cannot arise for
finite index inclusions by index considerations, or in the case when B ⊂ M is a masa. To establish
(1.1) for irreducible subfactors, it was necessary to first establish the general form of a one-sided
normalizer of a tensor product of irreducible subfactors and then deduce the normalizer result
from this. The same procedure is necessary here, so we introduce the notion of an intertwiner to
study groupoid normalizers in a one-sided situation.
Definition 1.2. Given an inclusion B ⊆ M of von Neumann algebras satisfying B ∩ M ⊆ B,
define the collection GN
(1)
M (B) of intertwiners of B in M by
GN
(1)
M (B) =
v a partial isometry in M: vBv∗
⊆ B,v∗
v ∈ B
.
We will write GN(1)
(B) for GN
(1)
M (B) when there is no confusion about the underlying
algebra M. We use the superscript (1) to indicate that our intertwiners are one-sided, namely
that although vBv∗ ⊆ B, we are not guaranteed to have a containment v∗Bv ⊆ B. Note that
v ∈ GNM(B) if, and only if, both v and v∗ lie in GN
(1)
M (B). Note too that while the groupoid
normalizers form a groupoid, the intertwiners do not. Finally, the terminology intertwiner comes
from the fact that, under the hypothesis B ∩ M ⊆ B, these are exactly the partial isometries that
witness the embeddability of a corner of B into itself inside M in the sense of Popa’s intertwining
procedure for subalgebras from [14,15].
We obtain a similar commutation result to (1.2) for intertwiners. In fact our main theorem,
stated below, obtains more as it gives approximate forms for intertwiners and groupoid normal-
izers of tensor products.
Theorem 1.3. Let Bi ⊂ Mi be inclusions of finite von Neumann algebras with separable preduals
and with fixed faithful normal traces τi on Mi. Moreover, suppose that B
i ∩Mi ⊆ Bi for i = 1,2.
For v ∈ GN(1)
M1⊗M2
(B1 ⊗ B2) and ε 0, there exist k ∈ N and operators x1,...,xk ∈ B1 ⊗ B2,
intertwiners w1,1,...,w1,k of B1 in M1 and intertwiners w2,1,...,w2,k of B2 in M2 such that](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-26-320.jpg)
![24 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
uniqueness of product measures on σ-finite spaces [17, p. 312]. The canonical trace on the basic
construction need not be a semifinite weight on Z(B
i ∩ Mi,eBi ) but does have this property
on the compression Z(B
i ∩ Mi,eBi )Qi where it can be treated as a measure (see Lemma 2.6
and the discussion preceding Definition 4.4). The remaining difficulty is to check that the pro-
jection Pv satisfies the hypotheses of Theorem 4.1, for which we require certain order properties
of the pull down map on the basic construction. These are described in the next section, in which
we also set out our notation, review the properties of the basic construction, and establish some
technical lemmas. Finally, the paper ends with Section 5, which handles the additional details
required to deduce the groupoid normalizer result (Theorem 5.5) from our earlier work.
2. Notation and preliminaries
Throughout the paper, all von Neumann algebras are assumed to have separable preduals.
The basic object of study in this paper is an inclusion B ⊆ M of finite von Neumann algebras,
where M is equipped with a faithful normal trace τ satisfying τ(1) = 1. We always assume that
M is standardly represented on the Hilbert space L2(M,τ), or simply L2(M). The letter ξ is
reserved for the image of 1 ∈ M in this Hilbert space, and J will denote the isometric conjugate
linear operator on L2(M) defined on Mξ by J(xξ) = x∗ξ, x ∈ M, and extended by continuity
to L2(M) from this dense subspace. Then L2(B) is a closed subspace of L2(M), and eB denotes
the projection of L2(M) onto L2(B), called the Jones projection. The von Neumann algebra
generated by M and eB is called the basic construction and is denoted by M,eB [2,9]. Let EB
denote the unique trace preserving conditional expectation of M onto B. In the next proposition
we collect together standard properties of eB,EB and M,eB from [9,13,10,18].
Proposition 2.1.
(i) eB(xξ) = EB(x)ξ, x ∈ M.
(ii) eBxeB = EB(x)eB = eBEB(x), x ∈ M.
(iii) M ∩ {eB} = B.
(iv) M,eB
= JBJ , Z( M,eB ) = JZ(B)J .
(v) eB has central support 1 in M,eB .
(vi) Span{xeBy: x,y ∈ M} generates a ∗-strongly dense subalgebra, denoted MeBM, of
M,eB .
(vii) x → eBx and x → xeB are injective maps for x ∈ M.
(viii) MeB and eBM are ∗-strongly dense in M,eB eB and eB M,eB respectively.
(ix) eB M,eB eB = BeB = eBB.
(x) (MeBM) M,eB (MeBM) ⊆ MeBM.
(xi) There is a unique faithful normal semifinite trace Tr on M,eB satisfying
Tr(xeBy) = τ(xy), x,y ∈ M. (2.1)
This trace is given by the formula
Tr(t) =
∞
i=1
tJv∗
i ξ,Jv∗
i ξ
, t ∈ M,eB
+
, (2.2)
where the vi’s are partial isometries in M,eB satisfying
∞
i=1 v∗
i eBvi = 1.](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-28-320.jpg)
![J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 25
(xii) The algebra MeBM is · 2,Tr-dense in L2( M,eB ,Tr) and · 1,Tr-dense in
L1( M,eB ,Tr).
(xiii) Given inclusions Bi ⊂ Mi of finite von Neumann algebras for i = 1,2, the basic construc-
tion M1 ⊗ M2,eB1⊗B2
is isomorphic to M1,eB1 ⊗ M2,eB2 . Under this isomorphism,
the canonical trace Tr on M1 ⊗M2,eB1⊗B2
is given by Tr1 ⊗Tr2, where Tri is the canon-
ical trace on Mi,eBi .
(xiv) There is a well-defined map Ψ : MeBM → M, given by
Ψ (xeBy) = xy, x,y ∈ M. (2.3)
This is the pull down map of [13], where it was shown to extend to a contraction from
L1( M,eB ,Tr) to L1(M,τ).
Using part (xii) of the previous proposition, the equation
Tr (xeBy)z = τ (xy)z = τ Ψ (xeBy)z , x,y,z ∈ M, (2.4)
shows that Ψ is the pre-adjoint of the identity embedding M → M,eB and is, in particular,
positive. The basic properties of Ψ are set out in [13], but we will need more detailed informa-
tion on this map than is currently available in the literature. We devote much of this section to
obtaining further properties of Ψ , the main objective being to apply them in Lemma 4.5.
In the next three lemmas, the inclusion B ⊂ M is always of arbitrary finite von Neumann
algebras with a fixed faithful normalized normal trace τ on M, inducing the trace Tr on M,eB .
Lemma 2.2. Let x ∈ L1( M,eB )+ ∩ M,eB . If Ψ (x) ∈ L1(M) ∩ M, then Ψ (x) ⩾ x.
Proof. It suffices to show that
Ψ (x)yξ,yξ
⩾ xyξ,yξ , y ∈ M. (2.5)
The maximality argument, preceding [18, Lemma 4.3.4], to establish part (xi) of Proposition 2.1
can be easily modified to incorporate the requirement that v1 = 1. Thus there are vectors ξi =
Jv∗
i ξ ∈ L2(M) so that (2.2) becomes
Tr(t) =
∞
i=1
tξi,ξi , t ∈ M,eB
+
, (2.6)
where ξ1 = ξ. Now, for y ∈ M, we may use the M-modularity of Ψ to write
Ψ (x)yξ,yξ
= Ψ y∗
xy ξ,ξ
= τ Ψ y∗
xy = Tr y∗
xy . (2.7)
It follows from (2.6) and (2.7) that
Ψ (x)yξ,yξ
= xyξ,yξ +
∞
i=2
xyξi,yξi ⩾ xyξ,yξ , y ∈ M, (2.8)
establishing that Ψ (x) ⩾ x. 2](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-29-320.jpg)
![26 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
We now extend this result to tensor products. Let N be a semifinite von Neumann algebra
with a specified faithful normal semifinite trace TR. In [5], Effros and Ruan identified the pre-
dual of a tensor product of von Neumann algebras X and Y by (X ⊗ Y)∗ = X∗ ⊗op Y∗, the
operator space projective tensor product of the preduals. In the presence of traces, this identi-
fies L1(X ⊗ Y) with L1(X) ⊗op L1(Y), so I ⊗ Ψ is well defined, positive, and bounded from
L1(N ⊗ M,eB ,TR ⊗ Tr) to L1(N ⊗ M,TR ⊗ τ), being the pre-adjoint of the identity em-
bedding N ⊗ M → N ⊗ M,eB . Following [21, Chapter IX], we will always assume that N is
faithfully represented on L2(N,TR), for which span{y ∈ N: TR(y∗y) ∞} is a dense subspace.
Lemma 2.3. Let x ∈ L1(N ⊗ M,eB )+ ∩ (N ⊗ M,eB ). If (I ⊗ Ψ )(x) ∈ L1(N ⊗ M) ∩
(N ⊗ M), then (I ⊗ Ψ )(x) ⩾ x.
Proof. Suppose that the result is not true. Then we may find a finite projection p ∈ N, elements
yi ∈ pNp and zi ∈ M, 1 ⩽ i ⩽ k, so that
x − (I ⊗ Ψ )(x)
k
i=1
yi ⊗ ziξ,
k
i=1
yi ⊗ ziξ
0, (2.9)
since such sums
k
i=1 yi ⊗ ziξ are dense in L2(N,TR) ⊗2 L2(M,τ). Then the inequality
(I ⊗ Ψ ) (p ⊗ 1)x(p ⊗ 1) ⩾ (p ⊗ 1)x(p ⊗ 1) (2.10)
fails. The element on the left of (2.10) is (p ⊗ 1)((I ⊗ Ψ )(x))(p ⊗ 1), and so is bounded by hy-
pothesis. The restriction of I ⊗ Ψ to L1(pNp ⊗ M,eB ) is the pull down map for the inclusion
pNp ⊗ B ⊆ pNp ⊗ M of finite von Neumann algebras with basic construction pNp ⊗ M,eB .
The failure of (2.10) then contradicts Lemma 2.2 applied to this inclusion, establishing that
(1 ⊗ Ψ )(x) ⩾ x. 2
The next lemma completes our investigation of the order properties of pull down maps.
Lemma 2.4. If x ∈ L1(N ⊗ M,eB )+ is unbounded, then so also is (1 ⊗ Ψ )(x).
Proof. Suppose that (1 ⊗ Ψ )(x) is bounded. Following [21, Section IX.2], we may regard x as a
self-adjoint positive densely defined operator on L2(N ⊗ M,eB ). For n ⩾ 1, let pn ∈ N be the
spectral projection of x for the interval [0,n]. Then pnx ⩽ x, so (I ⊗ Ψ (pnx)) ⩽ (I ⊗ Ψ )(x),
since I ⊗ Ψ is the pre-adjoint of a positive map. In particular, I ⊗ Ψ (pnx) is bounded. By
Lemma 2.3 applied to pnx,
(I ⊗ Ψ )(x) ⩾ (I ⊗ Ψ )(pnx) ⩾ pnx. (2.11)
Since n ⩾ 1 was arbitrary, we conclude from (2.11) that x is bounded, a contradiction which
completes the proof. 2
We note for future reference that these results are equally valid for pull down maps of the
form Ψ ⊗ I, due to symmetry. These lemmas will be used in Section 4 to derive an important
inequality. The next lemma formulates exactly what will be needed.](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-30-320.jpg)
![J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 27
Lemma 2.5. Let Bi ⊆ Mi, i = 1,2, be inclusions of finite von Neumann algebras with pull down
maps Ψi. Let B ⊆ M be the inclusion B1 ⊗ B2 ⊆ M1 ⊗ M2. If x ∈ L1( M,eB )+ ∩ M,eB is
such that (Ψ1 ⊗ Ψ2)(x) ∈ L1(M) ∩ M and the inequality
(Ψ1 ⊗ Ψ2)(x) ⩽ 1 (2.12)
is satisfied, then (I ⊗Ψ2)(x) ∈ L1( M1,eB1 ⊗M2)+ ∩( M1,eB1 ⊗M2) and (I ⊗Ψ2)(x) ⩽ 1.
Proof. Using the isomorphism of Proposition 2.1(xiii), Ψ1 ⊗ Ψ2 is the pull down map for
M,eB . Since (Ψ1 ⊗ I)((I ⊗ Ψ2)(x)) = (Ψ1 ⊗ Ψ2)(x) is a bounded operator by hypothesis,
it follows from Lemma 2.4 that (I ⊗ Ψ2)(x) is also bounded in M1,eB1 ⊗ M2. Thus the three
operators x, (I ⊗ Ψ2)(x) and (Ψ1 ⊗ Ψ2)(x) are all bounded, and so we may apply Lemma 2.3
twice to the pull down maps I ⊗ Ψ2 and Ψ1 ⊗ I to obtain
(Ψ1 ⊗ Ψ2)(x) ⩾ (I ⊗ Ψ2)(x) ⩾ x. (2.13)
The result then follows from (2.13) and the hypothesis (2.12). 2
In the proof of Lemma 4.6, we will need the following fact regarding inclusions of finite von
Neumann algebras B ⊂ M with B ∩ M ⊆ B. Here, and elsewhere in the paper, we consider
inclusions induced by cut-downs. Recall that if Q ⊆ N is an inclusion of von Neumann algebras
and q is a projection in Q, then
Q
∩ N q = (qQq)
∩ (qNq), Z Q
∩ N q = Z (qQq)
∩ (qNq) , (2.14)
see, for example, [18, Section 5.4].
Lemma 2.6. Let B ⊆ M be a containment of finite von Neumann algebras such that B ∩M ⊆ B.
If p ∈ M is a nonzero projection, then there exists a nonzero projection q ∈ B which is equivalent
to a subprojection of p.
Observe that if M is a finite factor, then Lemma 2.6 is immediate. Our proof of Lemma 2.6 is
classical, proceeding by analysing the center-valued trace on P . Alternatively one can establish
the lemma by taking a direct integral over the center. Since we have been unable to find this fact
in the literature we give the details for completeness.
Proof of Lemma 2.6. Let denote the center-valued trace on M. We will make use of two
properties of from [12, Theorem 8.4.3]. The first is that p1 p2 if and only if (p1) ⩽ (p2),
and the second is that p1 ∼ p2 if and only if (p1) = (p2).
The hypothesis B ∩ M ⊆ B implies that B ∩ M = Z(B) and, in particular, that Z(M) ⊆
Z(B). For some sufficiently small c 0, the spectral projection z of (p) for the interval [c,1]
is nonzero, and (pz) ⩾ cz. Since Bz ⊆ Mz also satisfies the relative commutant hypothesis, it
suffices to prove the result under the additional restriction (p) ⩾ c1 for some constant c 0.
Let n ⩾ c−1 be any integer. Suppose that it is possible to find a nonzero projection q ∈ B and
an orthogonal set {q,p2,...,pn} of equivalent projections in M. The sum of these projections
has central trace equal to n (q) and is also bounded by 1, so that (q) ⩽ n−11 ⩽ c1. But then
q p and we are done. Thus we may assume that there is an absolute bound on the length of](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-31-320.jpg)
![28 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
any such set, and we may then choose one, {q1,p2,...,pn}, of maximal length. By cutting by
the central support of q1, we may assume that this central support is 1.
Now consider the inclusion q1Bq1 ⊆ q1Mq1, and note that
(q1Bq1)
∩ q1Mq1 = q1 B
∩ M = q1Z(B) = Z(q1Bq1) (2.15)
from (2.14). Let f1 and f2 be nonzero orthogonal projections in q1Bq1 and q1Mq1 respectively.
By the comparison theory of projections, there exists a projection z ∈ Z(q1Mq1) ⊆ Z(q1Bq1) so
that
zf1 zf2, (1 − z)f2 (1 − z)f1, (2.16)
the equivalence being taken in q1Mq1. Now zf1 ∈ q1Bq1 and is equivalent to a subprojec-
tion p0 of zf2 ⩽ q1. Then the pair zf1,p0 is equivalent to orthogonal pairs below each pi,
2 ⩽ i ⩽ n, which will contradict the maximal length of {q1,p2,...,pn} unless zf1 = 0. Sim-
ilarly (1 − z)f2 = 0. Thus f1 and f2 have orthogonal central supports in q1Bq1 and so [18,
Lemma 5.5.3] shows that q1Bq1 is abelian. Eq. (2.15) then shows that q1Bq1 is a masa in
q1Mq1, and so another application of [18, Lemma 5.5.3] shows that q1Mq1 is also abelian.
Thus q1Bq1 = q1Mq1.
Now the projection 1 − q1 − p2 − ··· − pn must be 0, otherwise it would have a nonzero
subprojection equivalent to a nonzero projection q̃1 ∈ q1Mq1 = q1Bq1, since q1 has central sup-
port 1, and q̃1 would lie in a set of n + 1 equivalent orthogonal projections. Thus q1,p2,...,pn
are abelian projections in M with sum 1, so M is isomorphic to L∞(Ω) ⊗ Mn for some measure
space Ω. Identify p and q1 with measurable Mn-valued functions. Since q1 is abelian, the rank
of q1(ω) is 1 almost everywhere, and the rank of p(ω) is at least 1 almost everywhere since
(p) ⩾ c1. Then q1 is equivalent to a subprojection of p since (q1) ⩽ (p). This completes
the proof. 2
We conclude this section with a brief explanation of an averaging technique in finite von
Neumann algebras which we will use subsequently. It has its origins in [2], but is also used
extensively in [14]. If η ∈ L2(M) and U is a group of unitaries in M then the vector can be
averaged over U. This is normally associated with amenable groups, but can be made to work in
this setting without this assumption. Form the · 2-norm closure K of
K = conv
uηu∗
: u ∈ U
.
There is a unique vector η̃ ∈ K of minimal norm, and uniqueness of η̃ implies that uη̃u∗ = η̃
for all u ∈ U. We refer to η̃ as the result of averaging η over U, and many variations of this are
possible. We give an example of this technique by establishing a technical result which will be
needed in the proof of Theorem 4.7.
Recall (Proposition 2.1(xii)) that MeBM is · 2,Tr-dense in L2( M,eB ,Tr). Consider a ∗-
subalgebra A which is strongly dense in M. If x,y ∈ M, then fix sequences {xn}∞
n=1, {yn}∞
n=1
from A converging strongly to x and y, respectively. Then
(x − xn)eB
2
2,Tr
= Tr eB(x − xn)∗
(x − xn)eB
= τ (x − xn)∗
(x − xn) = (x − xn)ξ,(x − xn)ξ
, (2.17)](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-32-320.jpg)
![30 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
for 1 ⩽ i ⩽ n. Then K ⊆
n
i=1 Ki ⊗ gi. By [18, Lemma 9.2.1] K and each Ki are closed in
their respective · 2-norms. If k ∈ K is the element of minimal · 2-norm in K then it may be
written as k =
n
i=1 ki ⊗ gi with ki ∈ Ki. Since k is invariant for the action of U(N1 ⊗ 1), we
see that
n
i=1
ukiu∗
− ki ⊗ gi = 0, u ∈ U(N1). (2.21)
The linear independence of the gi’s allows us to conclude that ukiu∗ = ki for 1 ⩽ i ⩽ n and
u ∈ U(N1). Thus ki ∈ N
1 ∩ M1,eB1 = Z(B
1 ∩ M1,eB1 ). The inequality (2.20) is preserved by
averaging in this manner over U(N1 ⊗1) so, replacing each fi by ki if necessary, we may assume
that fi ∈ Z(B
1 ∩ M1,eB1 ) for 1 ⩽ i ⩽ n. Now repeat this argument on the right, averaging over
U(1⊗N2), to replace the gi’s by elements of Z(B
2 ∩ M2,eB2 ). With these changes, (2.20) now
approximates z by a sum from
L2
Z B
1 ∩ M1,eB1 ,Tr1 ⊗2 L2
Z B
2 ∩ M2,eB2 ,Tr2
which proves the containment from left to right and establishes equality. 2
3. Projections in the basic construction
In this section, we relate intertwiners of a subalgebra to certain projections in the basic con-
struction. We consider a finite von Neumann algebra M and a von Neumann subalgebra B whose
unit will always coincide with that of M. For the most part, we will be interested in the condition
B ∩ M ⊆ B (equivalent to B ∩ M = Z(B)), but we will make this requirement explicit when it
is needed.
Lemma 3.1. Let B be a von Neumann subalgebra of a finite von Neumann algebra M and let
v ∈ GN(1)
(B).
1. Then v∗eBv is a projection in (B ∩ M,eB )v∗v.
2. Suppose q is a projection in B. Then v∗eBv lies in (B ∩ M,eB )q if, and only if, v∗v ∈
Z(B)q = Z(qBq).
Proof. 1. The element v∗eBv is positive in M,eB . Since vv∗ ∈ B and so commutes with eB,
the following calculation establishes that v∗eBv is a projection:
v∗
eBv
2
= v∗
eBvv∗
eBv = v∗
vv∗
eBv = v∗
eBv. (3.1)
For an arbitrary b ∈ v∗vBv∗v,
v∗
eBv b = v∗
eBvbv∗
v = v∗
vbv∗
eBv = v∗
vbv∗
vv∗
eBv = b v∗
eBv , (3.2)
where the second equality uses vbv∗ ∈ B to commute this element with eB. Thus (3.2) establishes
that v∗eBv ∈ (v∗vBv∗v) ∩ v∗v M,eB v∗v which is (B ∩ M,eB )v∗v (see (2.14)).](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-34-320.jpg)
![J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 31
2. Suppose now that v∗v ∈ Z(B)q = Z(qBq). It is immediate that Z(qBq) ⊆ Z((B ∩
M,eB )q), so we have a decomposition
B
∩ M,eB q = B
∩ M,eB v∗
v ⊕ B
∩ M,eB q − v∗
v . (3.3)
We have already shown that v∗eBv is in the first summand of (3.3) so must lie in (B ∩ M,eB )q.
Conversely, the hypothesis on v∗eBv implies that v∗eBv = v∗eBvq = qv∗eBv, so the pull
down map gives v∗v = v∗vq = qv∗v, showing that v∗v ∈ qBq. For each b ∈ B, v∗eBvqbq =
qbqv∗eBv. Applying the pull down map gives v∗vqbq = qbqv∗v and hence v∗v ∈ Z(qBq). 2
We now strengthen this lemma under the additional hypothesis that B ∩ M ⊆ B. Recall from
part (iv) of Proposition 2.1 that JZ(B)J = Z( M,eB ). When B ⊂ M is a finite index inclusion
of irreducible subfactors, Lemma 3.2 is contained in [13, Propositions 1.7(2) and 1.9]. The proof
follows the extension to infinite index inclusions of irreducible subfactors in [20, Lemma 3.3].
Lemma 3.2. Let B be a von Neumann subalgebra of a finite von Neumann algebra M and
suppose that B ∩ M ⊆ B. Let v ∈ GN(1)
(B). Then the projection v∗eBv is central in (B ∩
M,eB )v∗v.
Proof. Define two projections p,q ∈ B by p = v∗v and q = vv∗. Now consider an arbitrary
x ∈ (B ∩ M,eB )p = (pBp) ∩ p M,eB p. Then, for each b ∈ B,
vxv∗
vbv∗
= vxpbpv∗
= vpbpxv∗
= vbv∗
vxv∗
, (3.4)
showing that vxv∗ ∈ (vBv∗) ∩ (q M,eB q). We next prove that qeB is central in (vBv∗) ∩
(q M,eB q). It lies in this algebra by the previous calculation and Lemma 3.1, as qeB =
v(v∗eBv)v∗.
Take t ∈ (vBv∗) ∩ (q M,eB q) to be self-adjoint and let η be tξ ∈ L2(M). Now take a
sequence {xn}∞
n=1 from M converging in · 2-norm to tξ. Since t = qt = tq, we may assume
that the sequence {xn}∞
n=1 lies in qMq, otherwise replace it by {qxnq}∞
n=1.
For each u in the unitary group U(pBp),t commutes with vuv∗ and so
Jvuv∗
Jvuv∗
η = Jvuv∗
Jvuv∗
tξ = Jvuv∗
Jtvuv∗
ξ
= tJvuv∗
Jvuv∗
ξ = tvuv∗
vu∗
v∗
ξ
= tqξ = tξ = η, (3.5)
where the third equality holds because vuv∗ ∈ B so that Jvuv∗J ∈ ( M,eB ). For each n ⩾ 1
and each u ∈ U(pBp),
Jvuv∗
Jvuv∗
xnξ − η 2
= Jvuv∗
Jvuv∗
(xnξ − η) 2
⩽ xnξ − η 2, (3.6)
from (3.5). If we let yn be the element of qMq obtained by averaging xn over the unitary group
vU(pBp)v∗ ⊆ qBq, then (3.6) gives
ynξ − η 2 ⩽ xnξ − η 2, n ⩾ 1, (3.7)](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-35-320.jpg)
![J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 33
and so f vbv∗ = vbv∗f . Thus
f ∈ vBv∗
∩ vv∗
Bvv∗
. (3.13)
If b0 ∈ B is such that vv∗b0vv∗ commutes with vBv∗, then
vv∗
b0vv∗
vbv∗
= vbv∗
vv∗
b0vv∗
, b ∈ B. (3.14)
Multiply on the left by v∗ and on the right by v to obtain
v∗
b0vv∗
vbv∗
v = v∗
vbv∗
vv∗
b0v, b ∈ B. (3.15)
Thus
v∗
b0v ∈ v∗
vBv∗
v
∩ v∗
vMv∗
v = v∗
v B
∩ M v∗
v = v∗
vZ(B)v∗
v. (3.16)
Consequently, vv∗b0vv∗ ∈ vZ(B)v∗. It follows that (vBv∗) ∩ vv∗Bvv∗ ⊆ vZ(B)v∗, and so
there is a central projection p ∈ Z(B) so that f = vpv∗. We now have
q = v∗
f eBv = v∗
vpv∗
eBv = pv∗
eBv, (3.17)
as required. 2
In Section 4, we wish to use the projection v∗eBv to investigate an intertwiner v of a ten-
sor product B = B1 ⊗ B2 ⊂ M1 ⊗ M2 = M, where each B
i ∩ Mi ⊆ Bi. In conjunction with
Proposition 2.1(xiii), Tomita’s commutation theorem gives
B
∩ M,eB
∼
= B
1 ∩ M1,eB1 ⊗ B
2 ∩ M2,eB2 . (3.18)
By Lemma 3.2, such an intertwiner gives rise to a central projection v∗eBv in (B ∩ M,eB )v∗v.
Unfortunately, in general the projection v∗v will not factorize as an elementary tensor of projec-
tions b1 ⊗ b2, with bi ∈ Bi, and so the algebra (B ∩ M,eB )v∗v will not decompose as a tensor
product. This prevents us from applying tensor product techniques to the projection v∗eBv di-
rectly. However, standard von Neumann algebra theory (see, for example, [11]) gives a central
projection P ∈ B ∩ M,eB such that v∗eBv = Pv∗v. Since we need to ensure that the projec-
tion P fully reflects the properties of v, we cannot just appeal to the general theory to obtain P ,
so we give an explicit construction in Definition 3.5 below. The subsequent lemmas set out the
properties of P that we require later.
Definition 3.5. Let B ⊂ M be an inclusion of finite von Neumann algebras with B ∩ M ⊆ B
and let v ∈ GN
(1)
M (B). Let z ∈ Z(B) be the central support of v∗v. Define p0 to be v∗v, and
let {p0,p1,...} be a family of nonzero pairwise orthogonal projections in B which is max-
imal with respect to the requirements that pn ⩽ z and each pn is equivalent in B to a sub-
projection in B of p0. Since two projections in a von Neumann algebra with non-orthogonal
central supports have equivalent nonzero subprojections, maximality gives
n⩾0 pn = z. For
n ⩾ 1, choose partial isometries wn ∈ B so that w∗
nwn = qn ⩽ p0 and wnw∗
n = pn. Then define](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-37-320.jpg)
![J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 35
(iii) If f ∈ N(P), then it is clear that f P is a projection below P , since f commutes with P .
Conversely, consider a projection Q ⩽ P with Q ∈ M,eB . The introduction of partial sums
below is to circumvent some questions of convergence.
Define Pk =
k
i=1 v∗
i eBvi. Then limk→∞ Pk = P strongly, so PkQPk converges strongly to
PQP = Q. For m,n ⩾ 1, let bm,n ∈ B be the element such that bm,neB = eBvmQv∗
neB. Then
PkQPk =
k
m,n=1
v∗
meBvmQv∗
neBvn =
k
m,n=1
v∗
mbm,neBvn
=
k
m,n=1
v∗
mbm,neBvnv∗
nvn =
k
m,n=1
v∗
mbm,nvnv∗
neBvn. (3.22)
Now define xk ∈ pW∗(GN(1)
(B))p by xk =
k
m,n=1 v∗
mbm,nvn. The relations viv∗
j = 0 for i = j
allow us to verify that xkP = PkQPk, and consequently
xkP = Px∗
k , k ⩾ 1, (3.23)
since PkQPk is self-adjoint. Thus
∞
i=1
xkv∗
i eBvi =
∞
i=1
v∗
i eBvix∗
k . (3.24)
The sums in (3.24) converge in · 1-norm, so we may apply the pull down map to obtain
xkp = px∗
k . Since xk = xkp, we conclude that xk is self-adjoint. Thus, from (3.23), xk commutes
with P , and so lies in N(P). From above, xkP ⩾ 0 and xkP ⩽ 1, so xk ⩾ 0 and xk ⩽ 1
by (ii). Let f be a σ-weak accumulation point of the sequence {xk}∞
k=1. Then f ⩾ 0, f ⩽ 1,
and f ∈ N(P). Since xkP = PkQPk, we conclude that f P = Q. It now follows from (ii) that f
is a projection in N(P).
If P ∈ B ∩ M,eB , then the pull down map gives bp = pb for b ∈ B, so p ∈ Z(B). If
also Q ∈ B ∩ M,eB , then Q = f P and f ⩽ p. If b ∈ Bp then commutation with Q gives
(bf − f b)P = 0, so f ∈ B ∩ M = Z(B), by (i). Finally Q = f P =
i(vif )∗eB(vif ), so is of
the same form as P . 2
4. Intertwiners of tensor products
In this section we will prove one of our main results, the equality of W∗(GN(1)
(B1)) ⊗
W∗(GN(1)
(B2)) and W∗(GN(1)
(B1 ⊗B2)), where Bi ⊆ Mi, i = 1,2, are inclusions of finite von
Neumann algebras satisfying B
i ∩ Mi ⊆ Bi. The key theorem for achieving this is the following
one, which enables us to detect those central projections in corners of the relative commutant
of the basic construction which arise from intertwiners. It is inspired by [1, Proposition 2.7],
although is not a direct generalization of that result. For comparison, [1, Proposition 2.7] shows
that, in the case of a masa A, a projection P ∈ A ∩ M,eA which is subequivalent to eA dom-
inates an operator v∗eBv for some v ∈ GN(A). Example 4.3 below will show that such a result
will not hold in general without additional hypotheses.](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-39-320.jpg)
![J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 39
The second case is when A is an arbitrary abelian von Neumann algebra, and Φ is bounded,
so we may assume that Φ is a state on A since (4.1) is unaffected by scaling. The result now
follows from the first case by replacing the inclusion B ⊆ M by A ⊗ B ⊆ A ⊗ M. The trace on
A ⊗ M is Φ ⊗ τ and eA⊗B is 1 ⊗ eB, so the canonical trace on A ⊗ M,eA⊗B is Φ ⊗ Tr.
The last case is where Φ is a faithful normal semifinite weight. Then there is a family {fλ}λ∈Λ
of orthogonal projections in A with sum 1 such that Φ(fλ) ∞ for each λ ∈ Λ. If Φλ denotes
the restriction of Φ to Afλ, then Φλ is bounded. If we replace P , A, Φ and q by respectively
P(fλ ⊗ 1), Afλ, Φλ and q(fλ ⊗ 1), then we are in the second case. Thus there exists, for each
λ ∈ Λ, a partial isometry vλ ∈ GN
(1)
Afλ⊗M
(Afλ ⊗ B) so that P(fλ ⊗ 1) = v∗
λ(eAfλ⊗B)vλ. The
central support of vλ lies below fλ ⊗ 1 so we may define v ∈ GN
(1)
A⊗M
(A ⊗ B) by v =
λ∈Λ vλ,
and it is routine to check that P = v∗eBv. 2
Theorem 4.1 characterizes those projections in the basic construction which arise from inter-
twiners.
Corollary 4.2. Given an inclusion B ⊆ M of finite von Neumann algebras with B ∩ M ⊆ B and
a projection q ∈ B, a projection P ∈ (B ∩ M,eB )q is of the form v∗eBv for some intertwiner
v ∈ GN(1)
(B) if and only if P eB in M,eB and Tr(Pr) ⩽ τ(qr) for all projections r ∈
Z(B)q. Furthermore in this case the domain projection v∗v must lie in Z(B)q.
Proof. Taking A = C and Φ to be the identity in Theorem 4.1 shows that any projection P sat-
isfying the conditions of the corollary is of the form v∗eBv for some intertwiner v ∈ GN(1)
(B).
Lemma 3.1(ii) then shows that v∗v ∈ Z(B)q. Conversely, given an intertwiner v ∈ GN(1)
(B),
Lemma 3.1(ii) shows that v∗eBv ∈ (B ∩ M,eB )q precisely when v∗v ∈ Z(B)q. The other
two conditions of the corollary follow as v∗eBv ∼ vv∗eB ⩽ eB in M,eB , and for a projection
r ∈ Z(B)q, Tr(v∗eBvr) = τ(v∗vr) ⩽ τ(qr). 2
The tracial hypothesis (4.1) of Theorem 4.1 is an extra ingredient in this theorem as compared
with [1, Proposition 2.7]. The following example shows that Theorem 4.1 can fail without this
hypothesis.
Example 4.3. Let R be the hyperfinite II1 factor and fix an outer automorphism θ of period two.
Let M = M3 ⊗ R and let
B =
a 0 0
0 b c
0 θ(c) θ(b)
: a,b,c ∈ R
⊆ M.
Note that B ∼
= R ⊕ (R θ Z2). It is straightforward to verify that B ∩ M = Z(B) = Ce11 ⊕
C(e22 + e33) where {ei,j }3
i,j=1 are the matrix units. Let P ∈ M,eB be (
√
2e12)eB(
√
2e21).
Since EB(e22) = (e22 + e33)/2, P is a projection, and it is routine to verify that P commutes
with B. If there is a nonzero intertwiner v ∈ M such that v∗eBv ⩽ P , then v = ve11. Direct cal-
culation shows that v would then have the form we11 for some partial isometry w ∈ R, so v∗eBv
would be qe11eB for some projection q ∈ R. However, this nonzero projection is orthogonal to
P and so cannot lie under it. Thus the conclusion of Theorem 4.1 fails in this case. Note that
Tr(Pe11) = τ(2e11) = 2/3, while τ(e11) = 1/3, so the tracial hypothesis of Theorem 4.1 is not
satisfied.](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-43-320.jpg)
![44 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
and so (4.36) implies that
v −
k
j=1
xj (w1,j ⊗ w2,j )
2,τ
ε,
as required. 2
Corollary 4.8. Let Bi ⊆ Mi, i = 1,2, be inclusions of finite von Neumann algebras satisfying
B
i ∩ Mi ⊆ Bi. Then
W∗
GN(1)
(B1 ⊗ B2) = W∗
GN(1)
(B1) ⊗ W∗
GN(1)
(B2) . (4.38)
There are two extreme cases where the hypothesis B ∩ M ⊆ B is satisfied. The first is when
B is an irreducible subfactor where the result of Theorem 4.7 can be deduced from the stronger
results of [20]. The second is when B is a masa in M where Theorem 4.7 is already known [1].
The following example explains the preference given to intertwiners over unitary normalizers in
intermediate cases, even in a simple setting.
Example 4.9. Let M be a II1 factor, let p ∈ M be projection whose trace lies in (0,1/2),
and let B = pMp + (1 − p)M(1 − p). This subalgebra has no non-trivial unitary normaliz-
ers, essentially because τ(p) = τ(1 − p). However, the tensor product B ⊗ B ⊆ M ⊗ M does
have such normalizers because the compressions by p ⊗ (1 − p) and (1 − p) ⊗ p, which have
equal traces, are conjugate by a unitary normalizer u which is certainly outside B ⊗ B. Accord-
ing to Theorem 4.7, u can be obtained as the limit of finite sums of elementary tensors from
W∗(GN(1)
(B)) ⊗ W∗(GN(1)
(B)).
5. Groupoid normalizers of tensor products
In this section we return to the groupoid normalizers GNM(B), namely those v ∈ M such
that v,v∗ ∈ GN
(1)
M (B). Our objective in this section is to establish a corresponding version of
Theorem 4.7 for GN(B), and consequently we will assume throughout that any inclusion B ⊆ M
satisfies the relative commutant condition B ∩ M ⊆ B.
We will need to draw a sharp distinction between those intertwiners v that are groupoid nor-
malizers and those that are not, and so we introduce the following definition.
Definition 5.1. Say that v ∈ GN(1)
(B) is strictly one-sided if the only projection p ∈ Z(Bv∗v) =
Z(B)v∗v for which vp ∈ GN(B) is p = 0. When B is an irreducible subfactor of M then any
unitary u ∈ M satisfying uBu∗ B is a strictly one-sided intertwiner (see [20, Example 5.4] for
examples of such unitaries).
Given v ∈ GN(B), recall from Section 3 that there is a projection Pv ∈ Z(B ∩ M,eB ) such
that Pvv∗v = v∗eBv, and Pv has the form
Pv =
n⩾0
v∗
neBvn, (5.1)](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-48-320.jpg)
![46 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
Q = zu∗
eBu = 0 (5.9)
from (5.2). This proves the result. 2
We can use the preceding lemma to show that a strictly one-sided intertwiner v ∈ GN(1)
(B)
has the property that the only projection p ∈ Bv∗v for which vp ∈ GN(B) is p = 0. Indeed, take
such a projection p for which w = vp ∈ GN(B). Then Pvw∗w = w∗eBw so that Pv ⩾ Pw by
Remark 3.7. Lemma 5.2 then gives Pwv∗eBv = 0. Thus Pww∗eBw = w∗eBw = 0 and so w = 0.
Lemma 5.3. Let v ∈ GN(1)
(B). Then there exist orthogonal families of orthogonal projections
en,fn ∈ v∗vBv∗v such that
(en + fn) = v∗v, each ven is strictly one-sided, and each vfn lies
in GN(B).
Proof. Let {en} be a maximal orthogonal family of projections in Z(B)v∗v such that ven is
strictly one-sided, and set e =
en. Then choose a maximal family of orthogonal projections
{fn} ∈ Z(B)(v∗v − e) such that vfn ∈ GN(B). If
en +
fn = v∗v then the result is proved,
so consider the projection g = v∗v −
en −
fn ∈ Z(B)v∗v and suppose that g = 0. Then vg
cannot be strictly one-sided otherwise the maximality of {en} would be contradicted. Thus there
exists z ∈ Z(B) such that vgz is a nonzero element of GN(B). But this contradicts maximality
of {fn}, proving the result. 2
We now return to considering two containments Bi ⊆ Mi satisfying B
i ∩ Mi ⊆ Bi, and the
tensor product containment B = B1 ⊗ B2 ⊆ M1 ⊗ M2 = M. The next lemma is the key step
required to obtain a version of Theorem 4.7 for groupoid normalizers. We need a result from the
perturbation theory of finite von Neumann algebras. For any containment A ⊆ N, where N has
a specified trace τ, recall that N ⊂δ,τ A means that
sup
x − EA(x) 2
: x ∈ N, x ⩽ 1
⩽ δ
where · 2 is defined using the given trace τ. If τ is scaled by a constant λ, then ⊂δ,λτ is the same
as ⊂δ/
√
λ,τ . Then [16, Theorem 3.5] (see also [18, Theorem 10.3.5]), stated for normalized traces,
has the following general interpretation: if A ⊆ N and N ⊂δ,τ A for some δ (τ(1)/23)1/2, then
there exists a nonzero projection p ∈ Z(A ∩ N) such that Ap = pNp.
Lemma 5.4. Let v ∈ GN(1)
(B1) and w ∈ GN(1)
(B2). If v (or w) is strictly one-sided then v ⊗
w ∈ GN(1)
(B) is strictly one-sided.
Proof. Without loss of generality, suppose that v is strictly one-sided. Fix a nonzero projection
p ∈ Z(B)(v∗v ⊗ w∗w), and let τi be the faithful normalized normal trace on Mi ⊃ Bi. Given
ε 0, we may choose projections pi ∈ Z(B1)v∗v and qi ∈ Z(B2)w∗w, 1 ⩽ i ⩽ k, with p −
k
i=1 pi ⊗qi 2,τ ε and the pi’s orthogonal since these projections lie in abelian von Neumann
algebras. Here, · 2,τ is with respect to the normalized trace τ = τ1 ⊗ τ2 on M. Now
piB1pi = piv∗
vB1v∗
vpi ⊆ piv∗
B1vpi (5.10)
since vB1v∗ ⊆ B1. If it were true that piv∗B1vpi ⊂δ,τ1 piB1pi for some δ (τ(pi)/23)1/2, then
it would follow from [16, Theorem 3.5] that there exists a nonzero projection p
i ∈ (piB1pi) ∩](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-50-320.jpg)
![48 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
Proof. Consider v ∈ GNM1⊗M2
(B1 ⊗ B2). Following the proof of Theorem 4.7, given ε 0,
there exist elements vi,j ∈ GN(1)
(Bi), 1 ⩽ j ⩽ k, so that
Pv −
k
j=1
(v1,j ⊗ v2,j )∗
eB(v1,j ⊗ v2,j )
2,Tr
ε (5.18)
as in (4.34). Using Lemma 5.3, we may replace this sum with one of the form
(w1,j ⊗ w2,j )∗
eB(w1,j ⊗ w2,j ) + (x1,j ⊗ x2,j )∗
eB(x1,j ⊗ x2,j )
where the wi,j ’s are two-sided and, for each j, at least one of x1,j ,x2,j is strictly one-sided. By
Lemma 5.4, each x1,j ⊗ x2,j is strictly one-sided, so (x1,j ⊗ x2,j )∗eB(x1,j ⊗ x2,j )Pv = 0 by
Lemma 5.2. If we multiply on the right by Pv, then
Pv −
j
(w1,j ⊗ w2,j )∗
eB(w1,j ⊗ w2,j )Pv
2,τ
ε. (5.19)
Simple approximation allows us to obtain the same estimate for some finite subcollection of the
w1,j and w2,j , say w1,1,...,w1,k and w2,1,...,w2,k. We now continue to follow the proof of
Theorem 4.7 from (4.34) to obtain the required xj . 2
Just as in Section 4, the next corollary follows immediately.
Corollary 5.6. Let Bi ⊆ Mi, i = 1,2, be inclusions of finite von Neumann algebras satisfying
B
i ∩ Mi ⊆ Bi. Then
GNM1 (B1)
⊗ GNM2 (B2)
= GNM1⊗M2
(B1 ⊗ B2)
.
Acknowledgments
The work in this paper originated during the Workshop in Analysis and Probability, held at
Texas AM University during Summer 2007. It is a pleasure to express our thanks to both the
organizers of the workshop and to the NSF for providing financial support to the workshop.
References
[1] I. Chifan, On the normalizing algebra of a masa in a II1 factor, arXiv:math.OA/0606225, 2006.
[2] E. Christensen, Subalgebras of a finite algebra, Math. Ann. 243 (1979) 17–29.
[3] J. Dixmier, Sous-anneaux abéliens maximaux dans les facteurs de type fini, Ann. of Math. (2) 59 (1954) 279–286.
[4] H.A. Dye, On groups of measure preserving transformation, I, Amer. J. Math. 81 (1959) 119–159.
[5] E.G. Effros, Z.-J. Ruan, On approximation properties for operator spaces, Internat. J. Math. 1 (2) (1990) 163–187.
[6] J. Fang, On completely singular von Neumann subalgebras, Proc. Edinb. Math. Soc. (2) 52 (2009) 607–618.
[7] J. Feldman, C.C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, I, Trans. Amer.
Math. Soc. 234 (2) (1977) 289–324.
[8] J. Feldman, C.C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, II, Trans. Amer.
Math. Soc. 234 (2) (1977) 325–359.
[9] V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1) (1983) 1–25.](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-52-320.jpg)
![J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 49
[10] V. Jones, V.S. Sunder, Introduction to Subfactors, London Math. Soc. Lecture Note Ser., vol. 234, Cambridge
University Press, Cambridge, 1997.
[11] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. I, Grad. Stud. Math., vol. 15,
American Mathematical Society, Providence, RI, 1997, Elementary theory, reprint of the 1983 original.
[12] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. II, Grad. Stud. Math., vol. 16,
American Mathematical Society, Providence, RI, 1997, Advanced theory, corrected reprint of the 1986 original.
[13] M. Pimsner, S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1) (1986) 57–106.
[14] S. Popa, On a class of type II1 factors with Betti numbers invariants, Ann. of Math. (2) 163 (3) (2006) 809–899.
[15] S. Popa, Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I, Invent. Math. 165 (2006)
369–408.
[16] S. Popa, A.M. Sinclair, R.R. Smith, Perturbations of subalgebras of type II1 factors, J. Funct. Anal. 213 (2) (2004)
346–379.
[17] H.L. Royden, Real Analysis, third ed., Macmillan Publishing Company, New York, 1988.
[18] A.M. Sinclair, R.R. Smith, Finite von Neumann Algebras and Masas, London Math. Soc. Lecture Note Ser., vol. 351,
Cambridge University Press, Cambridge, 2008.
[19] A.M. Sinclair, R.R. Smith, S.A. White, A. Wiggins, Strong singularity of singular masas in II1 factors, Illinois J.
Math. 51 (4) (2007) 1077–1083.
[20] R.R. Smith, S.A. White, A. Wiggins, Normalizers of irreducible subfactors, J. Math. Anal. Appl. 352 (2) (2009)
684–695.
[21] M. Takesaki, Theory of Operator Algebras, II, Operator Algebras and Non-commutative Geometry, vol. VI, Ency-
clopaedia Math. Sci., vol. 125, Springer-Verlag, Berlin, 2003.](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-53-320.jpg)
![Journal of Functional Analysis 258 (2010) 50–66
www.elsevier.com/locate/jfa
Linear maps preserving the minimum and reduced
minimum moduli
A. Bourhim a,∗,1,2
, M. Burgos b,3
, V.S. Shulman c
a Département de Mathématiques et de Statistique, Université Laval, Québec (Québec), Canada G1K 7P4
b Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
c Department of Mathematics, Vologda State Technical University, Vologda, Russia
Received 16 September 2008; accepted 2 October 2009
Communicated by Paul Malliavin
Abstract
We describe linear maps from a C∗-algebra onto another one preserving different spectral quantities such
as the minimum modulus, the surjectivity modulus, and the reduced minimum modulus.
© 2009 Elsevier Inc. All rights reserved.
Keywords: C∗-algebras; Linear preserver problems; Minimum modulus; Surjectivity modulus; Reduced minimum
modulus
1. Introduction
In the last few decades, there has been a considerable interest in the so-called linear preserver
problems which concern the characterization of linear or additive maps on matrix algebras or
operator algebras or more generally on Banach algebras that leave invariant a certain function,
a certain subset, or a certain relation; see for instance the survey papers [8,13,21,22,29] and the
* Corresponding author.
E-mail addresses: bourhim@mat.ulaval.ca, abourhim@syr.edu (A. Bourhim), mariaburgos@ugr.es (M. Burgos),
shulman_v@yahoo.com (V.S. Shulman).
1 Current address: Department of Mathematics, Syracuse University, 215 Carnegie Building, Syracuse, NY 13244,
USA.
2 Supported by an adjunct professorship at Laval university.
3 Partially supported by the Junta de Andalucía PAI project FQM-3737 and the I+D MEC project MTM2007-65959.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.jfa.2009.10.003](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-54-320.jpg)
![although it was observed about this time that six loads of money reached
Valladolid for him. Charles V, from his retirement of Yuste, wrote to him,
May 18th, expressing surprise that he, the creature of imperial favor, should
hesitate to repay the benefits conferred, especially as he could have what
security he desired for the loan. This letter, with one from Juana, was
conveyed to him by Hernando de Ochoa, whose report to Charles, May
28th, of the interview, showed how little respect was felt for the man.
Ochoa reproached him with having promised to see what he could do, in
place of which he had gone into hiding at San Martin de la Fuente, fourteen
leagues from the court at Valladolid, where he had lain for two months,
hoping that the matter would blow over. “He said to me, before a
consecrated host, that the devils could fly away with him if ever he had
100,000 or 80,000, or 60,000, or 30,000 ducats, for he had always spent
much in charities and had made dotations amounting to 150,000.” Ochoa
pressed him hard; he admitted that his archbishopric, which he had held
since 1546, was worth 60,000 ducats a year and Ochoa showed that,
admitting his claims for charities and expenses, he had laid aside at least
30,000 a year “which you cannot possibly have spent, for you never have
any one to dine in your house and you do not accumulate silver plate, like
other gentlemen; all this is notorious, and the whole court knows it.... This
embarrassed him, but he repeated with great oaths that he had no money,
that it was not well thus to oppress prelates, nor would money thus obtained
be lucky for war; God would help the king and what would Christendom
say about it.” The honest Ochoa still urged him to return to the court and
save his honor, intimating that the king might take action that would be
highly unpleasant, but it was to no purpose. Valdés was obdurate and clung
resolutely to his shekels.[133]
Philip had sent instructions as to the treatment of recalcitrants—probably
relegating bishops to their sees and nobles to their estates—but there was
hesitation felt as to banishing Valdés from the court, although the continued
pressure of Charles and Juana only extorted a promise of fifty thousand
ducats. Yet it was desired to remove him and plans were tried to offer him a
pretext for going. In March, 1588, Juana ordered him to accompany the
body of Queen Juana la loca to Granada for interment, from which place he
could visit his Seville church; he made excuses but promised to go shortly.
Then, when she repeated the order, he offered many reasons for evading it,
including the heresies recently discovered in Seville and Murcia; the](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-56-320.jpg)
![translation of the body could wait until September and everybody, he said,
was trying to drive him from the court. She referred the matter to the Royal
Council, which decided that his excuses were insufficient and that, even if
the interment were postponed he could properly be ordered to reside in his
see.[134]
It was evident to Valdés that something was necessary to strengthen his
position and he skilfully utilized the discovery of a few Protestants in
Valladolid, of whom some were eminent clerics, like Augustin Cazalla and
Fray Domingo de Rojas, and others were persons of quality, like Luis de
Rojas and Doña Ana Enríquez. We shall have occasion to note hereafter the
extraordinary excitement caused by the revelation that Protestantism was
making inroads in court circles, the extent of which was readily
exaggerated, and it was stimulated and exploited by Valdés, who magnified
his zeal in combating the danger and conjured, at least for the moment, the
storm that was brewing. Philip wrote from Flanders, June 5, 1558, to send
him to his see without delay; if he still made excuses he was to be excluded
from the Council of State and this would answer until his approaching
return to Spain, when he would take whatever action was necessary. Ten
days later, on receiving letters from Valdés enumerating the prisoners and
describing the efforts made to avert the danger, he countermanded the
orders.[135] Still, this was only a respite; we chance to hear of a meeting of
the Council of State, in August or September, in which Juan de Vega
characterized as a great scandal the disobedience of a vassal to the royal
commands, in a matter so just as residence in his see, and he suggested that,
when the court moved, no quarters should be assigned to Valdés, to which
Archbishop Carranza replied that it was no wonder that the orders of the
king were unable to effect what the commandments of God and the Church
could not accomplish.[136]
Something further was necessary to render him indispensable—
something that could be prolonged indefinitely and if, at the same time, it
would afford substantial relief to the treasury, he might be forgiven the
niggardness that had resisted the appeals of the sovereign. He had for some
time been preparing a scheme for this, which was nothing less than the
prosecution of the Primate of the Spanish Church, the income of whose see
was rated at from 150,000 to 200,000 ducats. To measure the full audacity
of this it is necessary to appreciate the standing of Archbishop Carranza.](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-57-320.jpg)
![ARCHBISHOP
CARRANZA
Bartolomé de Carranza y Miranda was born in 1503. At
the age of 12 he entered the university of Alcalá; at 18 he
took the final vows of the Dominican Order and was sent
to study theology in the college of San Gregorio at Valladolid, where, in
1530, he was made professor of arts, in 1533 junior professor of theology
and, in 1534, chief professor as well as consultor of the tribunal of
Valladolid. In 1540 he was sent as representative of his Order to the General
Chapter held in Rome, where he distinguished himself and was honored
with the doctorate, while Paul III granted him a licence to read prohibited
heretic books. On his return to Spain his reputation was national; he was
largely employed by the Suprema in the censorship of books, especially of
foreign Bibles, while the Councils of Indies and Castile frequently
submitted intricate questions for his judgement. In 1542 he was offered the
see of Cuzco, esteemed the wealthiest in the colonies, when he replied that
he would willingly go to the Indies on the emperor’s service but not to
undertake the cure of souls.[137] On the convocation of the Council of
Trent, in 1545, Charles V selected him as one of the delegates and, during
his three years’ service there, he earned the reputation throughout
Christendom of a profound theologian. When, in 1548, Prince Philip went
to join his father in Flanders, they both offered him the position of
confessor which he declined, as he did the see of Canaries which was
tendered to him in 1550. In this latter year he was elected provincial of his
Order for Castile and, in 1551, he was sent to the second convocation of the
Council of Trent by Charles and also as the representative of Siliceo,
Archbishop of Toledo. As usual, he played a prominent part in the Council
and, after its hasty dissolution, he remained there for some time employed
in the duty of examining and condemning heretical books. In 1553 he
returned to his professorship at Valladolid and when, in 1554, Prince Philip
sailed for England to marry Queen Mary and restore the island to the unity
of the Church, he took Carranza with him as the fittest instrument for the
work.[138]
Carranza subsequently boasted that, during his three years’ stay in
England, he had burnt, reconciled, or driven from the land thirty thousand
heretics and had brought two million souls back to the Church. If we may
believe his admiring biographers he was the heart and soul of the Marian
persecution and Philip did nothing in religious matters without his advice.
When, in September, 1555, Philip rejoined his father in Flanders, he left](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-58-320.jpg)
![ARCHBISHOP
CARRANZA
Carranza as Mary’s religious adviser, in which capacity he remained until
1557. Regarded by the heretics as the chief cause of their sufferings he
barely escaped from repeated attempts on his life by poison or violence.
[139] It is true that English authorities of the period make little mention of
him, but the continued confidence of Philip is ample evidence that his
persecuting zeal was sufficient to satisfy that exacting monarch.
When, in 1557, Carranza rejoined Philip in Flanders he was probably
engrossed in the preparation and printing of his large work on the
Catechism, of which more hereafter, but he still found time to investigate
and impede the clandestine trade of sending heretic books to Spain.[140]
That he had completely won Philip’s esteem and confidence was seen when
Siliceo of Toledo died, May 1, 1557, and Philip appointed him as successor
in the archbishopric. He refused the splendid prize and suggested three men
as better fitted for the place. Philip persisted; he was going to a neighboring
convent to confess and commune prior to the opening of the campaign and
ordered Carranza to obey on his return. When he came back he sent the
presentation written in his own hand; Carranza yielded, but on condition
that, as the war with the pope would delay the issue of the bulls, the king in
the interval could make another selection. This effort to avoid the fatal gift
was fruitless. On his return from the campaign, Philip in an autograph letter
summoned him to fulfil his promise and made the appointment public. So
high was Carranza’s reputation that, when the presentation was laid before
the consistory in Rome, on December 6th, it was at once confirmed, without
observing the preconization, or the customary inquiry into the fitness of the
appointee, or a constitution which prohibited final action on the same day.
[141]
The elevation of a simple friar to the highest place in
the Spanish Church was a blow to numerous ambitions
that could scarce fail to arouse hostility. Valdés himself
was said to have aspirations for the position and to be bitterly disappointed.
Pedro de Castro, Bishop of Cuenca, had also cherished hopes and was eager
for revenge. Carranza, moreover, was not popular with the hierarchy. He
was that unwelcome character, a reformer within the Church and, while
everyone acknowledged the necessity of reform, no one looked with favor
on a reformer who assailed his profitable abuses. As far back as 1547, while
in attendance on the Council of Trent, Carranza had preached a sermon on](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-59-320.jpg)
![one of the most crying evils of the time, the non-residence of bishops and
beneficiaries, and had embodied his views in a tractate as severe as a
Lutheran would have written on this abuse and the kindred one of
pluralities, to which possibly the stringent Tridentine provisions on the
subject may be attributed.[142] Such an outburst was not calculated to win
favor, seeing that the splendor of the curia was largely supported by the
prelacies and benefices showered upon its members and that in Spain there
was scarce an inquisitor or a fiscal who was not a non-resident beneficiary
of some preferment.
Carranza had, moreover, a peculiarly dangerous enemy in a brother
Dominican, Melchor Cano, perhaps the leading Spanish theologian of the
time when Spanish theology was beginning to dominate the Church.
Learned, able, keen-witted and not particularly scrupulous, he was in
intellect vastly superior to Carranza; there had been early rivalry, when both
were professors of theology, and causes of strife in the internal politics of
the Order had arisen, so that Cano could scarce view without bitterness the
sudden elevation of his brother fraile.[143] His position at the time was
somewhat precarious. When, in 1556, Paul IV forced war on Philip II, that
pious prince sought the advice of theologians as to the propriety of
engaging in hostilities with the Vicegerent of God, and the parecer, or
opinion which Cano drew up, was an able state paper that attracted wide
attention. He defended uncompromisingly the royal prerogatives, he
virtually justified the German revolt when the Centum Gravamina of the
Diet of Nürnberg, in 1522, were unredressed and he described the
corruption of Rome as a disease of such long standing as to be incurable.
[144] This hardy defiance irritated Paul in the highest degree. April 21, 1556
he issued a brief summoning that son of perdition, Melchor Cano, to appear
before him within sixty days for trial and sentence, but the brief was
suppressed by the Royal Council and Cano was ordered not to leave the
kingdom. The Spanish Dominicans rallied to his defence; in the chapter of
1558 he was elected provincial and deputy to the general chapter to be held
in Rome, but Paul ordered the election to be annulled and Cano to be
deprived of his priorate of San Esteban. Cano complained of lukewarmness
in his defence on the part of both Philip and Carranza and it is easy to
understand that, feeling keenly the disgrace inflicted on him, he was in a
temper to attack any one more fortunate than himself.[145]](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-60-320.jpg)
![ARCHBISHOP
CARRANZA
At this inauspicious moment Carranza presented
himself as a fair object of attack by all who, from different
motives, might desire to assail him. If we may judge from
his writings, he must have been impulsive and inconsiderate in his speech,
given to uttering extreme views which made an impression and then
qualifying them with restrictions that were forgotten. He was earnestly
desirous of restoring the Church to its ancient purity and by no means
reticent in exposing its weaknesses and corruption. He had been trained at a
time before the Tridentine definitions had settled points of faith which,
since the twelfth century, had been the subjects of debate in the schools, and
even in his maturity the Council of Trent had not yet been clothed with the
awful authority subsequently accorded to it, for the inglorious exit of its
first two convocations, in 1547 and 1552, gave little promise of what lay in
the future. The echo of the fierce Lutheran controversies had scarce
penetrated into Spain and comparatively little was there known of the
debates which were shaking to its centre the venerable structure of the
Church. Carranza’s very labors in condemning heretic books and converting
heretics had acquainted him with their doctrines and modes of expression;
he was a confused thinker and his impulsive utterances were liable to be
construed in a sense which he did not anticipate. As early as 1530 he had
been denounced to the Inquisition by Fray Juan de Villamartin as a defender
of Erasmus, especially in the matter of confession and the authorship of the
Apocalypse and, during his persecuting career in England, he more than
once gave opportunity, in his sermons, to unfavorable comment.[146] It was
also in evidence that when in Rome, in 1539, he had written to Juan de
Valdés in Naples, asking what authors should be studied for understanding
Scripture, as he would have to teach that subject, and that Valdés replied in
a letter which Carranza circulated among his students in Valladolid—a letter
highly heretical in its teachings which Valdés subsequently included in his
“One hundred and ten Divine Considerations.”[147] It is true that, in 1539,
Juan de Valdés was not reckoned a heretic, but, if the letter was correctly
identified with the “Consideration” in question its circulation was highly
imprudent, for it asserted that the guides for the study of Scripture are
prayer inspired by God and meditation based on spiritual experience, thus
discarding tradition for private interpretation, and it further dwelt upon the
confidence which the soul should feel in justification through Christ. In the
death-struggle with Protestantism the time had passed for easy-going](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-61-320.jpg)
![latitude of opinion and, in the intricate mazes of scholastic theology, it was
necessary to walk warily, for acute censorship could discover heresy in any
unguarded expression. The great services rendered by Cardinal Morone and
Cardinal Pole did not save them from the prosecuting zeal of Paul IV and
Contarini and Sadoleto were both suspect of heresy.[148] Under such
conditions a rambling inconsequential thinker like Carranza was peculiarly
open to attack.
He had unquestionably been more or less intimate with some of the
prominent personages whose arrest for Lutheranism, in the spring of 1558,
produced so immense a sensation. It was not unnatural that, on their trials,
they should seek to shield themselves behind his honored name, but the
detached fragments of conversation which were cited in support of vague
general assertions, even if correctly reported, amount to nothing in the face
of the emphatic testimony by Fray Domingo de Rojas, for the discharge of
his conscience, a few hours before his execution, that he had never seen in
Carranza anything that was not Catholic in regard to the Roman Church and
all its councils, definitions and laws and that when Lutherans were alluded
to he said their opinions were crafty and deceiving; they had sprung from
hell and the incautious could easily be deceived by them.[149] The credence
due to the evidence of the Lutherans, on which so much stress was laid, can
be gauged by a subsequent case illustrative of the tendency to render
Carranza responsible for all aberrations of belief. A certain Gil Tibobil (de
Bonneville) on trial in 1564 for Lutheranism, in Toledo, sought to palliate
his guilt by asserting that he had heard Carranza preach, in the church of
San Agustin, against candles and images and that confession was to be
made to God and not to the priest. This was too crude to be accepted and he
was sternly told that it cast doubt on the rest of his confession for, if
Carranza had thus preached publicly, it would have come to the knowledge
of the Inquisition and he would have been punished.[150]
Whether the testimony acquired in the trials of the Lutherans was
important or not, Inquisitor-general Valdés lost no time in using it to
discredit Carranza in the opinion of the sovereigns. As early as May 12,
1588, in a report to Charles V at Yuste, his assistance is asked in obtaining
the arrest of a fugitive, whose capture would be exceedingly important; he
had been traced to Castro de Urdiales, where he was to embark for Flanders
to find refuge with Carranza or with his companion Fray Juan de](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-62-320.jpg)
![ARCHBISHOP
CARRANZA
Villagarcia, where he was sure of being well received. That the real motive
was to injure Carranza with Charles appears from Valdés repeating the story
to him in a report of June 2, adding that the fugitive had escaped and that
information had been sent to Philip in order that he might be captured.[151]
It is reasonable to assume that whatever incriminating evidence could be
obtained from the prisoners was promptly brought to the notice of the
sovereigns and that inferences were unscrupulously asserted as facts.
At this critical juncture, Carranza delivered himself into
the hands of his enemies. In England and Flanders he had
employed the intervals of persecution in composing a work
which should set forth the irrefragable truths of the Catholic faith and guard
the people from the insidious poison of heretical doctrine. This was a task
for which, at such a time, he was peculiarly unfitted. He was not only a
loose thinker but a looser writer, diffuse, rambling and discursive, setting
down whatever idea chanced to occur to him and wandering off to whatever
subjects the idea might suggest. Moreover he was earnest as a reformer
within the Church, realizing abuses and exposing them fearlessly—in fact,
he declared in the Prologue that his object was to restore the purity and
soundness of the primitive Church, which was precisely what the heretics
professed as their aim and precisely what the ruling hierarchy most dreaded.
[152] Worst of all, he did this in the vulgar tongue, unmindful of the extreme
reserve which sought to keep from the people all knowledge of the errors
and arguments of the heretics and of the contrast between apostolic
simplicity and the splendid sacerdotalism of a wealthy and worldly
establishment.[153] This he cast into the form of Commentaries on the
Catechism, occupying a folio of nine hundred pages, full of impulsive
assertions which, taken by themselves, were of dangerous import, but which
were qualified or limited, or contradicted in the next sentence, or the next
page, or, perhaps, in the following section.
No one, I think, can dispassionately examine the Commentaries without
reaching the conviction that Carranza was a sincere and zealous Catholic,
however reckless may seem many of his isolated utterances. Nor was his
orthodoxy merely academic. He belonged to the Church Militant and his
hatred of heresy and heretics breaks out continually, in season and out of
season, whether apposite or not to his immediate subject. Heretic arguments
are not worthy of confutation—it is enough to say that a doctrine is](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-63-320.jpg)
![condemned by the Church and therefore it is heretical. The first duty of the
king is to preserve his dominions in the true faith and to chastise those who
sin against it. Even if heretics should perform miracles, their disorderly
lives and corrupted morals would be sufficient to guard the people from
listening to them or believing them. If they do not admit their errors they
are to be condemned to death; this is the best theology that a Christian can
learn and it was not more necessary in the time of Moses than it is at
present.[154]
Even in that age, when theology was so favorite a topic, few could be
expected to wade through so enormous a mass of confused thinking and
disjointed writing, and it was easy for Carranza’s enemies to garble isolated
sentences by which he could be represented to the sovereigns as being at
least suspect in the faith, and suspicion of heresy was quite sufficient to
require prosecution. Carranza himself, after his book was printed, seems to
have felt apprehension and to have proceeded cautiously in giving it to the
public. A set of the sheets was sent to the Marchioness of Alcañizes and a
dozen or more copies were allowed to reach Spain, where they were
received in March, 1558. Pedro de Castro, Bishop of Cuenca, obtained one
and speedily wrote to Valdés, denouncing the writer as guilty of heretical
opinions. Valdés grasped the opportunity and ordered Melchor Cano to
examine the work. Cano took as a colleague Fray Domingo de Cuevas and
had no difficulty in discovering a hundred and one passages of heretical
import. The preliminaries to a formal trial were now fairly under way, the
result of which could scarce be doubtful under inquisitorial methods, if the
royal and papal assent could be obtained, necessary even to the Inquisition
before it could openly attack the Primate of the Spanish Church.
Despite the profound secrecy enveloping the operations of the
Inquisition, it was impossible that, in an affair of such moment, there should
not be indiscretions and Carranza in Flanders was advised of what was on
foot. His friends urged him not to return to Spain but to take refuge in Rome
under papal protection, but he knew that this would irrevocably cost him the
favor of Philip, for exaggerated jealousy of papal interference with the
Inquisition was traditional since the time of Ferdinand and Isabella, and he
virtually surrendered his case at once by instructing his printer, Martin
Nucio, not to sell copies of the Commentaries without his express orders,
thus withdrawing it from circulation.[155]](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-64-320.jpg)
![ARCHBISHOP
CARRANZA
But little adverse impression seems as yet to have been
made on Philip. When Carranza was about to leave
Flanders, the king gave him detailed instructions which
manifest unbounded confidence. He was to go directly to Valladolid and
represent the extreme need of money; then he was to see Queen Mary of
Hungary, Charles’ sister, and persuade her to come to Flanders; then he was
to hasten to Yuste where Philip, through him, unbosomed himself to his
father, revealing all his necessities and desires in family as well as in state
affairs. In short, Carranza was still one whom he could safely entrust with
his most secret thoughts.[156]
Carranza, with his customary lack of worldly wisdom, threw away all
the advantages of his position. Landing at Laredo on August 1st, he passed
through Burgos, where he was involved in an unseemly squabble with the
archbishop over his assumed right to carry his archiepiscopal cross in
public. He did not reach Valladolid until the 13th and there he tarried,
busied ostensibly with a suit between his see and the Marquis of Camarasa
over the valuable Adelantamiento of Cazorla, but doubtless occupied also
with efforts to counteract the intrigues of Valdés. Then he performed his
mission to Mary of Hungary and it was not until the middle of September
that he set out on a leisurely journey to Yuste. Valdés had taken care to
forestall his visit. An autograph letter of the Princess Juana to Charles,
August 8th, says that Valdés had asked her to warn him to be cautious in
dealing with Carranza, for he had been implicated by the Lutheran prisoners
and would already have been arrested had he been anyone else. Charles was
naturally impatient to see him, not only to obtain explanations as to this, but
also to receive the messages expected from Philip, for which he was waiting
before writing to Flanders. Carranza’s delay, in spite of repeated urgency
from Yuste, could not but create a sinister impression and all chance of
justification was lost, for Charles was prostrated by his fatal illness before
Carranza left Valladolid and the end was near when he reached Yuste about
noon on September 20th. Charles expired the next morning at half-past two,
Carranza administering to him the last consolations, his method in which
formed one of the charges against him on his trial. He had thrown away his
last chance and the unexpected death of Charles deprived him of one who
might possibly have stood between him and his fate.[157]](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-65-320.jpg)
![ARCHBISHOP
CARRANZA
The plans of Valdés were now sufficiently advanced for him to seek the
papal authorization which alone was lacking, and his method to obtain this
was characteristically insidious. The Suprema addressed, September 9th, to
Paul IV a relation of its labors in discovering and prosecuting the Lutheran
heretics. There was skilful exaggeration of the danger impending from a
movement, the extent of which could not be known, and it was pointed out
that sympathy with the sectaries might be entertained by officials of the
Inquisition itself, by the Ordinaries and the consultors; so that extraordinary
powers were asked to arrest and judge and relax those suspected or guilty,
even though they were persons holding a secular or pontifical and
ecclesiastical dignity or belonging to any religious or other Order.[158] As
the Inquisition already had jurisdiction over all but bishops (it had not
hesitated to arrest and try the Dominican Fray Domingo de Rojas) the self-
evident object of this was to obtain surreptitiously, under cover of the word
“pontifical,” some general expression that might be used to deprive
Carranza of his right to trial by the pope. The Dean of Oviedo, a nephew of
Valdés, was sent to Rome as a special agent to procure the desired brief;
whether royal sanction for this application was obtained does not appear,
but it probably was not, at least at this stage.
Carranza meanwhile had been vainly endeavoring to get
copies of the censures on his book in order to answer them.
He appealed earnestly to his friends in Philip’s court and in
Rome but, without awaiting their replies, he pursued his policy of
submission and, on September 21st, the day of Charles’s death, he wrote to
Sancho López de Otálora, a member of the Suprema, that he consented to
the prohibition of his work, provided this was confined to Spain and that his
name was not mentioned.[159] In this and what followed he has been
accused of weakness, but it is difficult to see what other course lay open to
him. He doubtless still considered his episcopal consecration a guarantee
for his personal safety, while his reputation for orthodoxy could best be
conserved by not entering into a fruitless contest with a power irresistible in
its chosen field of action—a contest, moreover, which would have cost him
the royal favor that was his main reliance.
In pursuance of this policy he even descended to attempting to propitiate
Melchor Cano by offering to do whatever he would recommend. Cano
subsequently asserted, with customary mendacity, that Carranza would have](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-66-320.jpg)
![averted his fate had he adopted any of the means which he devised and
advised to save him, but it is difficult to imagine what more he could have
done.[160] Towards the close of November he wrote to Valdés and the
Suprema and to other influential persons professing his submission. He
explained the reasons which had led him to write his book in the vernacular
after commencing it in Latin; it could readily be suppressed for, on reaching
Valladolid, he had withdrawn the edition from the printer; there were no
copies in the bookshops and what he had brought with him he would
surrender, while the dozen or so that had been sent to Spain could easily be
called in as the recipients were all known. Then, on December 9th, he
proposed to the Suprema that the book should be prohibited in Spanish and
be returned to him for correction and translation into Latin.[161] Had the
real object of Valdés been the ostensible one of preserving the faith, this
would have amply sufficed; the book would have been suppressed and the
public humiliation of the Archbishop of Toledo, so distinguished for his
services to religion, would have been an amply deterrent warning to all
indiscreet theologians. It was a not unnatural burst of indignation when, in a
letter to Domingo de Soto, November 14th, he bitterly pointed out how the
heretics would rejoice to know that Fray Bartolomé de Miranda was treated
in Spain as he had treated them in England and Flanders and that, after he
had burnt them to enforce the doctrines of his book, it was pronounced in
Spain unfit to be read.[162] Carranza’s submission brought no result save to
encourage his enemies, who put him off with vague replies while awaiting
the success of their application to the pope.
Meanwhile he had reached Toledo, October 13th, and had applied
himself actively to his duties. He was rigid in the performance of divine
service, he visited prisons, hospitals and convents, he put an end to the sale
of offices and charging fees for licences, he revised the fee-bill of his court,
he enforced the residence of parish priests and was especially careful in the
distribution of preferment—in short he was a practical as well as theoretical
reformer. His charity also was boundless, for he used to say that all he
needed was a Dominican habit and that whatever God gave him was for the
poor. Thus during his ten months of incumbency, he distributed more than
eighty thousand ducats in marrying orphans, redeeming captives,
supporting widows, sending students to universities and in gifts to hospitals.
[163] He was a model bishop, and the resolute fidelity with which the](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-67-320.jpg)
![ARCHBISHOP
CARRANZA
chapter of Toledo supported his cause to the end shows the impression
made on a body which, in Spanish churches, was usually at odds with its
prelate.
He had likewise not been idle in obtaining favorable opinions of his
book from theologians of distinction. In view of the rumors of inquisitorial
action, there was risk in praising it, yet nearly all those prominent in
Spanish theology bore testimony in its favor. The general view accorded
virtually with that of Pedro Guerrero, Archbishop of Granada, than whom
no one in the Spanish hierarchy stood higher for learning and piety. The
book, he said, was without error and, being in Castilian, was especially
useful for parish priests unfamiliar with Latin, wherefore it should be
extensively circulated. It was true that there were occasional expressions
which, taken by themselves, might on their face seem to be erroneous, but
elsewhere it was seen that they must be construed in a Catholic sense. To
this effect recorded themselves Domingo and Pedro de Soto, men of the
highest reputation, Garrionero Bishop of Almería, Blanco of Orense, Cuesta
of Leon, Delgado of Lugo and numerous others.[164] If some of these men
belied themselves subsequently and aided in giving the finishing blow to
their persecuted brother, we can estimate the pressure brought to bear on
them.
Valdés speedily utilized the power of the Inquisition to
check these appreciations of the Commentaries. When, at
the University of Alcalá, the rector, the chancellor, and
twenty-two doctors united in declaring the work to be without error or
suspicion of error, save that some incautious expressions, disconnected
from the context, might be mistaken by hasty readers, Valdés muzzled it and
all other learned bodies and individuals by a letter saying that it had come
to his notice that learned men of the university had been examining books
and giving their opinions. As this produced confusion and contradiction
respecting the Index which the Inquisition was preparing, all persons,
colleges and universities were forbidden to censure or give an opinion
concerning any book without first submitting it to the Suprema, and this
under pain of excommunication and a fine of two hundred ducats on each
and every one concerned.[165] It was impossible to contend with an
adversary armed with such weapons. Not content with this, the rector of the
university, Diego Sobaños, was prosecuted by the tribunal of Valladolid for](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-68-320.jpg)
![the part he had taken in the matter; he was reprimanded, fined and absolved
ad cautelam. Similar action was taken against the more prominent of those
who had expressed themselves favorably and who, for the most part, were
forced to retract.[166] The Inquisition played with loaded dice.
Dean Valdés of Oviedo meanwhile had succeeded in his mission to
Rome, aided, as Raynaldus assures us, by the express request of Philip,
though this is more than doubtful. The brief was dated January 7, 1559; it
was addressed to Valdés and recited that, as there were in Spain some
prelates suspected of Lutheranism, he was empowered for two years from
the receipt of the brief, with the advice of the Suprema, to make
investigation and, if sufficient proof were found against any one and there
was good reason to apprehend his flight, to arrest and keep him in safe
custody, but as soon as possible the pope was to be informed of it and the
prisoner was to be sent to him with all the evidence and papers in the case.
[167] With the exception of the provision against expected flight, this was
merely in accordance with the received practice in the case of bishops, but
it was the entering wedge and we shall see how its limitations were
disregarded.
The brief was received April 8th. In place of complying with it and
sending Carranza to Rome with the evidence that had been collecting for
nearly a year, a formal trial was secretly commenced. The fiscal presented a
clamosa or indictment, on May 6th, asking for Carranza’s arrest and the
sequestration of his property, “for having preached, written and dogmatized
many errors of Luther.” The evidence was duly laid before calificadores, or
censors, who reported accordingly and, on the 13th, there was drawn up a
summons to appear and answer to the demand of the fiscal. Before
proceeding further, in an affair of such magnitude, it was felt that the assent
was required of Philip, who was still in Flanders.[168] As recently as April
4th he had replied encouragingly to an appeal from the persecuted prelate.
“I have not wanted to go forward in the matter of your book, about which
you wrote to me, until the person whom you were sending should arrive; he
has spoken with me today. I had already done something of what is proper
in this business. Not to detain the courier who goes with the good news of
the conclusion of peace, I do not wish to enlarge in replying to you, but I
shall do so shortly and meanwhile I earnestly ask you to make no change in
what you have done hitherto and to have recourse to no one but to me, for it](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-69-320.jpg)
![ARCHBISHOP
CARRANZA
would be in the highest degree disadvantageous.”[169] Philip evidently
thought that only Carranza’s book and not his person was concerned, that
the affair was of no great importance and his solicitude was chiefly to
prevent any appeal to Rome, a matter in which he fully shared the intense
feeling of his predecessors. When Carranza ordered his envoy to Flanders,
Fray Hernando de San Ambrosio, to proceed to Rome and secure an
approbation of the Commentaries, he replied, April 19th, that all his friends
at the court earnestly counselled against; it had been necessary to assure
Philip of the falsity of the reports that he had done so, whereupon the king
had expressed his satisfaction and had said that any other course would
have displeased him.[170]
Advantage, for which Carranza foolishly offered the
opportunity, was taken of this extreme jealousy to win him
over. When the Dominican chapter met, in April, 1559,
there was open strife between him and Cano, over a report that Cano had
styled him a greater heretic than Luther and that he favored Cazalla and the
other prisoners. Carranza demanded his punishment for the slander and
sought to defeat his candidacy for the provincialate. In this he failed. Cano’s
assertion that he had been misunderstood was accepted; he was again
elected provincial and Carranza unwisely carried his complaint to Rome.
[171] There it became mixed up with the question of Cano’s confirmation,
for Paul IV naturally resented the repeated presentation of that “son of
iniquity.” Philip, on the other hand, could not abandon the protection of one
whose fault, in papal eyes, was his vindication of the royal prerogative, and
he interested himself actively in pressing the confirmation. Paul
equivocated and lied and sought some subterfuge which was found in
Cano’s consecration, in 1552, as Bishop of Canaries (a post which he had
resigned in 1553) which was held to render him ineligible to any position in
his Order, and a general decree to that effect was issued in July.[172]
All this was skilfully used to prejudice Philip against Carranza. In letters
of May 16th to him and of May 22nd and 25th to his confessor Bernardo de
Fresneda, Cano with great adroitness and small respect for veracity
represented himself as subjected to severe persecution. He had always been
Carranza’s friend; he had withheld for seven months his censure of the
Commentaries and had yielded only to a threat of excommunication and
now Carranza was repaying him by intriguing against the confirmation in](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-70-320.jpg)
![ARCHBISHOP
CARRANZA
Rome—the truth being that it was not until the end of June that Carranza’s
agent reached there. It was a terrible thing, Cano added, if the archbishop,
through his Italian General, could thus wrong him and he could not defend
himself. He was resolved to suffer in silence, but the persecution was so
bitter that if the king did not speedily come to Spain he would have to seek
refuge in Flanders.[173] What, in reality, were his sufferings and what the
friendly work on which he was engaged, are indicated by a commission
issued to him, May 29th, granting him the extraordinary powers of a
substitute inquisitor-general and sending him forth on a roving expedition to
gather evidence, compelling everyone whom he might summon to answer
whatever questions he might ask.[174] The Suprema and Valdés, moreover,
in letters of May 13th and 16th to Philip, adopted the same tone; Cano’s
labors throughout the affair had been great and it was hoped that the king
would not permit his persecution for the services rendered to God and his
majesty; there need be no fear of injustice to Carranza, for the investigation
was impartial and dispassionate.[175]
Philip had already been informed by Cardinal Pacheco, February 24th
and again May 13th, that Carranza had sent to the pope copies of the
favorable opinions of his book, asking that it be judged in Rome and that
his episcopal privilege of papal jurisdiction be preserved.[176] Whatever
intentions he had of befriending Carranza were not proof against the
assertions that to his intrigues was attributable the papal interference with
Cano’s election. On June 26th he wrote to Cano, expressing his satisfaction
and assuring him of his support in Rome and, on the same day, to the
Suprema approving its actions as to the Commentaries and expressing his
confidence that it would do what was right.[177] In thus authorizing the
prosecution he ordered the archbishop’s dignity to be respected and he
wrote to the Princess Juana that, to avoid scandal, she should invite him to
Valladolid to consult on important matters, so that the trial could proceed
without attracting attention.[178]
Philip’s letters were received July 10th, but there was
still hesitation and it was not until August 3d that the
princess wrote, summoning Carranza in haste to
Valladolid, where she would have lodgings prepared for him. This she sent,
with secret instructions, by the hands of Rodrigo de Castro, a member of the
Suprema.[179] Carranza was at Alcalá de Henares, whither Diego Ramírez,](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-71-320.jpg)
![inquisitor of Toledo, was also despatched, under pretext of publishing the
Edict of Faith. Carranza, who suspected a snare, was desirous of postponing
his arrival at Valladolid until Philip, on whose protection he still relied,
should reach Spain. Accordingly he converted the journey into a visitation,
leaving Alcalá on the 16th and passing through Fuente el Saz and
Talamanca to Torrelaguna, which he reached on the 20th. On the road he
received intimations of what was in store and at Torrelaguna Fray Pedro de
Soto came with the news that emissaries had already started to arrest him,
which elicited from him a despairing and beseeching letter to Fresneda, the
royal confessor.[180]
De Soto’s report was true. Valdés dreaded as much as Carranza desired
Philip’s arrival; the delay on the road risked this if the device of the
invitation to Valladolid was to be carried out. For his plans it was essential
that an irrevocable step should be taken in the king’s absence—a step which
should compromise Carranza and commit the Inquisition so fully that Philip
could not revoke it without damaging the Holy Office in a way that to him
was impossible. To allow Carranza to be at liberty while investigating the
suspicion of his heresy, as Philip had ordered, would leave the door open to
royal or papal intervention; to seize and imprison him would leave Philip no
alternative but to urge forward his destruction, while his dilatory progress
could be assumed to cover preparations for flight. Accordingly, on August
17th the Suprema issued a commission, under the papal brief of January
7th, to Rodrigo de Castro to act with other inquisitors in the case, while, as
justice required Carranza’s arrest, Valdés commissioned de Castro, Diego
Ramírez and Diego González, inquisitor of Valladolid, to seize the person
of the archbishop and convey him to such prison as should be designated, at
the same time sequestrating all his property, real and personal and all his
papers and writings. Simultaneously Joan Cebrian, alguazil mayor of the
Suprema, was ordered to coöperate with the inquisitors in the arrest and
sequestration.[181]
Cebrian started the same day for Torrelaguna, where he kept his bed
through the day and worked at night. The inquisitors came together; a force
of familiars and others was secretly collected and, by day-break on the 22nd
the governor, the alcalde and the alguaziles of Torrelaguna were seized and
held under guard, the house where Carranza lodged was surrounded, de
Castro, Ramírez, Cebrian and a dozen men ascended the stairs and knocked](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-72-320.jpg)
![ARCHBISHOP
CARRANZA
at the door of the antechamber. Fray Antonio de Utrilla asked who was
there and the dread response came “Open to the Holy Office!” It was the
same at the door of Carranza’s chamber; de Castro knelt at the bed-side,
where Carranza had drawn the curtains and raised himself on his elbow; he
begged Carranza’s pardon with tears in his eyes and said his face would
show his reluctance in performing his duty. Cebrian was called in and read
the order of arrest. Carranza replied “These señores do not know that they
are not my judges, as I am subject directly to the pope.” Then de Castro
produced the papal brief from the bosom of his gown and read it. Some say
that Carranza fell back on his pillow, others that he remained imperturbable.
He ordered out all the rest and remained for a considerable time alone with
de Castro and Ramírez.[182]
He was at once secluded in the most rigid manner, all his people being
excluded, except Fray Domingo Ximenes, who was required to assist in the
sequestration and inventory. At table he was served by de Castro and
Ramírez, who treated him with the utmost respect and endeavored to
console him, for by this time his fortitude had given way and he was
overwhelmed. His attendants were all dismissed and given money to find
their way whither they chose and their grief we are told moved every one to
compassion. Only the cook and steward and a muleteer were retained to
serve the party. At nine in the evening proclamation was made throughout
the town that until daylight no one was to leave his house or look out of a
window. At midnight Cebrian assembled forty horsemen; de Castro and
Ramírez brought Carranza down and stationed themselves on either side of
his mule as the cavalcade rode forth in the darkness and then Salinas, the
owner of the house, was allowed to come out to close his door. The heat
was overpowering and when, by ten in the morning they reached Lozoya,
they rested for a day and a night. On the 27th they arrived at Laguna del
Duero, near Valladolid, where de Castro and Ramírez left the party and rode
forward for instructions, returning the same day and, at two in the morning
of the 28th, Carranza was brought to the city and lodged in the house of
Pedro González de Leon, in the suburb of San Pedro beyond the walls,
which had been taken by the Inquisition.[183]
Carranza thus disappeared from human sight as
completely as though swallowed by the earth. It is a
forcible illustration of inquisitorial methods, but](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-73-320.jpg)
![conspicuous only by reason of the dignity of the victim, for it rested with
the discretion of the officials whether thus to spirit away and conceal their
prisoners or to cast them publicly into the secret prison. Morales tells us
that it was years before the place of Carranza’s incarceration was known,
although every one said that he had been seized by the Holy Office. Even to
say this, however, was not unattended with danger, for, in the trial, in
September, by the tribunal of Toledo, of Rodrigo Alvárez, one of the
charges against him was that, about September 5th, he had remarked to a
casual fellow-traveller, that he came from Valladolid and was quite certain
that the archbishop was imprisoned.[184]
There could be no doubt about it in Toledo, where the news of the arrest
was received on the 24th. On the 26th the chapter assembled in sorrow to
take what measures they could, in aid of their beloved prelate, but they were
powerless save to delegate two of their number to reside in Valladolid and
render such assistance as was possible. It amounted to little save a
testimony of sympathy, for no communication was allowed, but they
advised with his counsel and performed what service they were able. This
faithful watch was kept up during the long and weary years of the trial and
when it was adjourned to Rome they went thither and remained to the end.
The chapter also, almost monthly, sent memorials to Philip praying for a
speedy and favorable end of the case. The great Dominican Order also felt
keenly the disgrace inflicted on its distinguished member and exerted itself
in his favor as far as it could. The Spanish episcopate also was greatly
perturbed, not knowing where the next blow might fall and the scandal
throughout the land was general.[185]
Philip had disembarked at Laredo on August 29th. Valdés evidently felt
that some excuse was necessary for action so much more decisive than that
prescribed by the king and, in a letter of September 9th, explained to him
that Carranza was delaying his movements in order to meet him on his
arrival at Laredo; that he was working in Rome to impede the matter; that
the infamy of his position was daily spreading and that the auto de fe
prepared for the Lutherans could not take place while he was at liberty.
Seeing that the effort to entice him to Valladolid had failed, it was resolved
to bring him there, which was done quietly and without disturbance. He had
been well treated and would continue to be so and the king might rely on
the affair being conducted with all rectitude. An intimation, moreover, that](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-74-320.jpg)
![ARCHBISHOP
CARRANZA
all his property had been sequestrated indicates that the financial aspect of
the matter was deemed worthy of being called to the royal attention and the
whole tone of the letter shows that Carranza’s imprisonment was
predetermined. The allusion to his design of meeting the king at Laredo
disposes of the plea that he was suspected of flight and the fact that the auto
de fe of the Lutherans did not take place until October 8th is a test of the
flimsiness of the reasons alleged.[186]
Carranza’s treatment was vastly better than that of ordinary prisoners
confined in the cells of the secret prison. He was asked to select his
attendants, when he named six, but was allowed only two—his companion,
Fray Alonso de Utrilla and his page, Jorje Gómez Muñoz de Carrascosa.
[187] Two rooms were allotted to the party—rooms without provision for the
needs of human nature, with windows padlocked and shutters closed, so
that at times the stench became unendurable. The foul atmosphere brought
on a dangerous illness in which Carranza nearly perished; the physicians
ordered the apartment to be ventilated, morning and evening, but all that the
Suprema would permit was a small grating in the door, though at times it
was left ajar with a guard posted at it.[188] Communication with the outside
world was so completely cut off that when, in 1561, a great conflagration
ravaged Valladolid, raging for thirty hours, destroying four hundred houses
and penetrating to the quarter where the prison stood, the prisoners knew
nothing of it until after reaching Rome.[189] The inquisitorial rule that all
consultation with counsel must be held in the presence of an inquisitor was
rigidly observed and also that which denied to prisoners the consolation of
the sacraments.
Diego González, one of the inquisitors of Valladolid,
was assigned to the special charge of Carranza who, in a
long and rambling memorial to the Suprema represents
him as treating him without respect, insulting him, suppressing his
communications with the Suprema, fabricating answers, throwing every
impediment in the way of his defence and aggravating, with malicious
ingenuity, the miseries of his position. Some details as to the parsimony
with which he was treated are almost incredible when we reflect that the
Inquisition and Philip were enjoying the enormous sequestrated revenues of
their prisoner.[190]](https://image.slidesharecdn.com/1220652-250606143059-860a90bb/85/Journal-Of-Functional-Analysis-Volume-258-Issues-1-2-3-4-A-Connes-75-320.jpg)
Journal Of Functional Analysis Volume 258 Issues 1 2 3 4 A Connes
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- 5. Journal of Functional Analysis 258 (2010) 1–19 www.elsevier.com/locate/jfa On OL∞ structure of nuclear, quasidiagonal C∗-algebras Caleb Eckhardt Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, United States Received 25 June 2008; accepted 2 October 2009 Communicated by Alain Connes Abstract We continue the study of OL∞ structure of nuclear C∗-algebras initiated by Junge, Ozawa and Ruan. In particular, we prove if OL∞(A) < 1.005, then A has a separating family of irreducible, stably fi- nite representations. As an application we give examples of nuclear, quasidiagonal C∗-algebras A with OL∞(A) > 1. © 2009 Elsevier Inc. All rights reserved. Keywords: Operator spaces; C∗-algebras; Quasidiagonal; Nuclear C∗-algebras 1. Introduction This paper continues the study of OL∞-structure of nuclear C∗-algebras initiated by Junge, Ozawa and Ruan in [8]. Before describing the contents of this paper, we recall the necessary definitions and results. Let V and W be n-dimensional operator spaces and consider the completely bounded version of Banach–Mazur distance: dcb(V,W) = inf ϕcb ϕ−1 cb : ϕ : V → W is a linear isomorphism . Let A be a C∗-algebra. For λ 1 we say that OL∞(A) ⩽ λ if for every finite-dimensional subspace E ⊂ A, there exist a finite-dimensional C∗-algebra B and a subspace E ⊂ F ⊂ A such E-mail address: ceckhard@uiuc.edu. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.004
- 6. 2 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 that dcb(F,B) ⩽ λ. Then define OL∞(A) = inf λ: OL∞(A) ⩽ λ . A is a rigid OL∞ space if for every 0 and every x1,...,xn ∈ A there is a finite-dimensional C∗-algebra B and a complete isometry ϕ : B → A such that dist(xi,ϕ(B)) for i = 1,...,n. OL∞ is an interesting invariant for C∗-algebras, particularly when one considers the interplay between OL∞ and various approximation properties of C∗-algebras. It follows easily from the definition, that if OL∞(A) ∞, then there is a net of matrix algebras (Mni ) and linear maps αi : A → Mni , βi : Mni → A such that βiαi tends to the identity on A pointwise and supi αicbβicb ∞. Pisier showed [11, Theorem 2.9] that this implies A is nuclear. Conversely, it was shown in [8] if A is nuclear, then OL∞(A) ⩽ 6. This estimate was improved in [7] when the authors showed that all nuclear C∗-algebras A have OL∞(A) ⩽ 3. So, OL∞ is most useful when restricted to nuclear C∗-algebras. Another important approximation property is quasidiagonality (QD). We refer the reader to the survey article [5] for information on QD C∗-algebras. The following relationships between QD and OL∞ were established in [8]: A is a rigid OL∞ space (i) − → OL∞(A) = 1 (ii) − − → A is nuclear QD. Blackadar and Kirchberg showed [3, Proposition 2.5] that all 3 of the above assertions are equiv- alent if A is either simple or both prime and antiliminal. The main purpose of this paper is to give examples showing that the converse of (ii) does not hold in general. In Section 2 we prove the necessary technical results used throughout the paper. Section 3 contains our first counterexamples to (ii). Section 4 contains some results about permanence properties about OL∞. In Section 5 we prove the main result that all unital C∗-algebras A with OL∞(A) 1.005 have a separating family of irreducible, stably finite representations. This provides a larger class of nuclear quasidiagonal C∗-algebras A with OL∞(A) 1, but also has implications for the converse of (i) which we discuss at the end of the paper. 2. Technical lemmas In this section, we gather some technical lemmas needed for Sections 3 and 4, and fix our notation. Throughout the paper, if H is a Hilbert space, we let B(H) denote the space of bounded linear operators on H. For H n-dimensional we write 2(n), and Mn for B(2(n)). We write ucp and cpc as shorthand for “unital completely positive” and “completely positive contraction” respectively. For linear maps ϕ : V → W between operator spaces we write ϕ(n) for idMn ⊗ ϕ :Mn(V ) → Mn(W), and ϕcb = supnϕ(n). Furthermore if ϕ is injective, we write ϕ−1 for the norm of the map ϕ−1 : ϕ(V ) → V . We write ⊗ for the minimal tensor product of C∗- algebras. The following lemma is implicit in the proof of [8, Theorem 3.2]. Lemma 2.1. Let 0 δ 1/ √ 2, and let A be a unital C∗-algebra with OL∞(A) 1 + δ2/2. Let F ⊂ A be a finite subset. Then there is a finite-dimensional C∗-algebra B, a linear map ϕ : B → A with ϕcb 1 + δ2/2 and a ucp map ψ : A → B such that F ⊂ ϕ(B) and ψϕ − idBcb 1 + δ2 /2 2 δ2 + δ4/4 .
- 7. C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 3 In [8, Theorem 3.2] the authors require δ 1/16. The reason for this is to guarantee that ψ is approximately multiplicative on F . We will not need approximate multiplicativity in this paper, which is why we are able to relax this condition to δ 1/ √ 2. We need the following slight variation of Lemma 2.1. Lemma 2.2. Let λ (1+ √ 3 2 )1/2 and A be a unital C∗-algebra with OL∞(A) λ. Let F ⊂ A be a finite subset. Then there is a finite-dimensional C∗-algebra B, a ucp map ψ : A → B and a unital, self-adjoint map ϕ : B → A such that: (i) ϕcb λ 1−λ √ 2(λ2−1) . (ii) F ⊂ ϕ(B). (iii) ψϕ = idB. (iv) ϕψ|F = idF . Proof. Without loss of generality suppose F consists of positive elements. We apply Lemma 2.1 with λ = 1 + δ2/2 to obtain a finite-dimensional C∗-algebra B, a ucp map ψ : A → B, and a linear map ϕ : B → A such that F ⊂ ϕ(B), ϕcb λ and ψϕ − idBcb λ 2(λ2 − 1) 1. Then ψϕ is invertible in the Banach algebra of all completely bounded maps on B. Let ϕ = ϕ(ψϕ)−1. Then ϕ cb ⩽ ϕcb (ψϕ)−1 cb ⩽ λ 1 1 − λ 2(λ2 − 1) . Then ϕ satisfies (i)–(iii). Moreover, since ψ is unital and ψϕ = idB, it follows that ϕ is unital. Finally, let ϕ (x) = 1/2(ϕ (x) + ϕ (x∗)∗), for x ∈ B. Then ϕ is unital, self-adjoint and ϕ cb ⩽ ϕ cb. Since ψ is positive, it follows that ψϕ = idB. To see (ii), let b ∈ B such that ϕ (b) ∈ F . Since F consists of positive elements, b = ψϕ (b) ⩾ 0. Hence, ϕ (b) = 1/2(ϕ (b) + ϕ (b)∗) = ϕ (b) ∈ F . Condition (iv) is a consequence of (ii) and (iii). 2 Lemma 2.3. Let A be a unital C∗-algebra and let x ∈ A. Set x1 = x1 x x∗ x1 ∈ M2 ⊗ A. (2.1) Then x1 = 2x. Proof. Without loss of generality, assume that x = 1. Clearly x1 ⩽ 2. For the reverse in- equality, suppose that A ⊂ B(H) unitally for some Hilbert space H. By spectral theory there is a sequence of unit vectors (ηk) ⊂ H such that lim k→∞ x∗ xηk − ηk = 0. (2.2) For each k ∈ N set ξk = 1 √ 2 xηk ηk ∈ H ⊕ H.
- 8. 4 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 Then ξk ⩽ 1 and by (2.2), it follows that lim k→∞ x1ξk = lim k→∞ 1 √ 2 2xηk x∗xηk + ηk = 2. Hence, x1 ⩾ 2. 2 Lemma 2.4. Let A and B be C∗-algebras with A unital, and 1/2 r ⩽ 1. Let ϕ : A → B be a cpc such that for every k ∈ N and a ∈ Mk ⊗ A with a ⩾ 0, ϕ(k)(a) ⩾ ra. Then ϕ is injective with ϕ−1cb ⩽ (2r − 1)−1. Proof. Let n ∈ N and x ∈ Mn ⊗ A. Let x1 ∈ M2 ⊗ (Mn ⊗ A) be as in Lemma 2.3. Then, x1 ⩾ 0 and x1 = 2x. By assumption, we have 2rx ⩽ ϕ(2n) (x1) = xϕ(n)(1) ϕ(n)(x) ϕ(n)(x)∗ xϕ(n)(1) ⩽ x1 ϕ(n)(x) ϕ(n)(x)∗ x1 ⩽ x + ϕ(n) (x) . Hence ϕ(n)(x) ⩾ (2r − 1)x, from which we conclude that ϕ−1 cb ⩽ (2r − 1)−1 . 2 We recall the following well-known corollary to Stinespring’s Theorem. Lemma 2.5. Let A and B be unital C∗-algebras and ψ : A → B a ucp map. Then for every a ∈ A, we have ψ(a)∗ψ(a) ⩽ ψ(a∗a). Lemma 2.6. Let L1 and L2 be Hilbert spaces and n ∈ N. Let ϕ : Mn → B(L1) ⊕ B(L2) be an injective cpc with ϕ−1cb = r−1 2/( √ 6 − 1). Let ϕi : Mn → B(Li) denote the coordinate maps of ϕ for i = 1,2. Suppose there is a k ∈ N and a ∈ Mk ⊗ Mn of norm 1 and a ⩾ 0 such that ϕ (k) 2 (a) = s r2 + r − 1 /r. Then ϕ1 is injective and ϕ−1 1 cb ⩽ r − 1 − r2 1 − s −1 . (2.3)
- 9. C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 5 Proof. By [13, Theorem 2.10] every bounded map ψ from an operator space into Mn is com- pletely bounded with ψ(n) = ψcb. So, we may assume that k = n. Also by [13, Theo- rem 2.10], to prove inequality (2.3) it suffices to show that for every x ∈ Mn ⊗ Mn of norm 1, we have ϕ (n) 1 (x) ⩾ r − 1 − r2 1 − s . (2.4) By Wittstock’s extension theorem [9, Theorem 8.2], let ψ : B(L1) ⊕ B(L2) → Mn be an extension of ϕ−1 : ϕ(Mn) → Mn with ψcb = ϕ−1cb = r−1. Let ψ = r ψ. Then ψcb = 1 and ψϕ(x) = rx for all x ∈ Mn. (2.5) By the factorization theorem for completely bounded maps [9, Theorem 8.4] there is a unital representation (π,H) of Mn ⊗ B(L1) ⊕ B(L2) = B L1 ⊗ 2 (n) ⊕ B L2 ⊗ 2 (n) and isometries S,T : 2(n) ⊗ 2(n) → H such that T ∗ π(x)S = ψ(n) (x) for every x ∈ B L1 ⊗ 2 (n) ⊕ B L2 ⊗ 2 (n) . (2.6) Let qL1 = π(1L1⊗2(n),0) ∈ B(H) and qL2 = π(0,1L2⊗2(n)) ∈ B(H). We now show that the ranges of S and T are almost included in qL1 (H). Let ξ1 ∈ 2(n) ⊗ 2(n) be a norm 1 eigenvector for a with eigenvalue 1. Let ω1 ∈ Mn ⊗ Mn be the orthogonal projection onto Cξ1. Then ω1 ⩽ a. Since ϕ2 is cp, we have ϕ (n) 2 (ω1) ⩽ ϕ (n) 2 (a) = s. Extend ξ1 to an orthonormal basis ξ1,ξ2,...,ξn2 for 2(n) ⊗ 2(n). For i = 1,...,n2 define the rank 1 operators, ωi(η) = η,ξi ξ1, for η ∈ 2 (n) ⊗ 2 (n). Then ωiω∗ j = δi,j ω1, for 1 ⩽ i,j ⩽ n2 . (2.7) Let η = n2 i=1 αiξi ∈ 2(n) ⊗ 2(n) of norm 1 and ωη = n2 i=1 αiωi ∈ Mn ⊗ Mn. By (2.7) and Lemma 2.5, it follows that ϕ (n) 2 (ωη) = ϕ (n) 2 (ωη)ϕ (n) 2 (ωη)∗ 1/2 ⩽ n2 i=1 |αi|2 ϕ (n) 2 (ω1) 1/2 ⩽ s1/2 . (2.8)
- 10. 6 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 Combining (2.5) and (2.6), we have rξ1 = rωη(η) = ψ(n) ◦ ϕ(n) (ωη)η = T ∗ π ϕ(n) (ωη) Sη. Therefore, by (2.8) r2 ⩽ π ϕ(n) (ωη) Sη 2 = π ϕ (n) 1 (ωη),0 qL1 Sη 2 + π 0,ϕ (n) 2 (ωη) qL2 Sη 2 ⩽ qL1 Sη2 + sqL2 Sη2 . (2.9) Combining (2.9) with the fact that S is an isometry, we obtain 1 = qL1 Sη2 + qL2 Sη2 ⩾ r2 − sqL2 Sη2 + qL2 Sη2 . Since η ∈ 2(n) ⊗ 2(n) was an arbitrary vector of norm 1, it follows that qL2 S ⩽ 1 − r2 1 − s 1/2 . (2.10) Define ψ∗ : B(L1) ⊕ B(L2) → Mn by ψ∗(x) = ψ(x∗)∗. By the complete positivity of ϕ it fol- lows that ψ∗ϕ = r · idMn . Moreover note that ψ∗ (n) (x) = S∗ π(x)T. So, by replacing ψ with ψ∗ (and hence S with T ) in the above proof we obtain T ∗ qL2 = qL2 T ⩽ 1 − r2 1 − s 1/2 . (2.11) Let x ∈ Mn ⊗ Mn be arbitrary of norm 1. By (2.5), (2.6), then (2.10) and (2.11), we have r = ψ(n) ϕ(n) (x) = T ∗ π ϕ(n) (x) S = T ∗ qL1 π ϕ (n) 1 (x),0 qL1 + qL2 π 0,ϕ (n) 2 (x) qL2 S ⩽ ϕ (n) 1 (x) + T ∗ qL2 π 0,ϕ (n) 2 (x) qL2 S ⩽ ϕ (n) 1 (x) + 1 − r2 1 − s . This proves (2.4) and the lemma. 2
- 11. C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 7 We will be careful with our norm estimates throughout the paper. Thus, the technical nature of Lemma 2.6. Colloquially it states; regardless of the value of n ∈ N, if ϕ is almost a complete isometry, then either ϕ1 or ϕ2 is almost a complete isometry. In particular, we have: Corollary 2.7. Let L1,L2,n and ϕ be as in Lemma 2.6, but with ϕ−1cb = r−1 125/124. Then either ϕ1 or ϕ2 is injective, and ϕ−1 i cb ⩽ 1 + (r − 1)1/3 −1 for either i = 1 or i = 2. Proof. If ϕ2 is injective with ϕ−1 2 cb (1 + (r − 1)1/3)−1, we are done. If not, then there is an x ∈ Mn ⊗ Mn of norm 1 such that ϕ (n) 2 (x) 1 + (r − 1)1/3. Then Lemma 2.4 provides an a ∈ Mn ⊗ Mn of norm 1 with a ⩾ 0 such that ϕ (n) 2 (a) ⩽ 1 2 1 + ϕ (n) 2 (x) ⩽ 1 2 2 + (r − 1)1/3 . We now apply Lemma 2.6 with s = 1 2 (1 + ϕ (n) 2 (x)) to obtain, ϕ−1 1 cb ⩽ r − 1 − r2 1 − s −1 ⩽ 1 + (r − 1)1/3 −1 , which holds whenever 124/125 r ⩽ 1. 2 Finally, we recall 2 useful perturbation lemmas. Lemma 2.8. (See [16, Proposition 1.19].) Let A be a unital C∗-algebra and N an injective von Neumann algebra. Let ϕ : A → N be a unital self-adjoint map with ϕcb ⩽ 1 + for some 0. Then there is a ucp map t : A → N such that t − ϕcb ⩽ . Lemma 2.9. (See [12, Lemma 2.13.2].) Let 0 1 and X be an operator space. Let (xi, xi)n i=1 be a biorthogonal system with xi ∈ X and xi ∈ X∗. Let y1,...,yn ∈ X be such that xixi − yi . Then there is a complete isomorphism w : X → X such that w(yi) = xi and wcbw−1cb ⩽ 1+ 1− . 3. First examples For 1 ⩽ λ (1+ √ 3 2 )1/2, let f (λ) = λ 1 − λ √ 2(λ − 1) , (3.1)
- 12. 8 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 and consider the real polynomial, g(y) = y(1 + y)(y − 1)(2 − y) − 2(2 − y)2 + 1. (3.2) Note that f (λ) → 1 as λ → 1, and g(1) = −1. Let λ in the domain of f be such that g f (λ ) 0. (3.3) A calculation shows that any λ 1.005 satisfies (3.3). Theorem 3.1. Let A be a unital C∗-algebra and let λ satisfy (3.3). Suppose that A has a unital faithful representation (π,Hπ ) = (ρ ⊕ σ,Hρ ⊕ Hσ ), such that ker(σ) = {0}. Furthermore sup- pose there is a sequence (xn) in the unit sphere of A such that ρ(xn) is an isometry for each n, and ρ(xnx∗ n) → 0 strongly in B(Hρ). Then OL∞(A) ⩾ λ . Proof. Let a ∈ ker(σ) be positive and norm 1. Choose n large enough so ρ(1−xnx∗ n)ρ(a)ρ(1− xnx∗ n) = 0. Set y = xn, and let b = 1 − yy∗ a 1 − yy∗ −1 1 − yy∗ a 1 − yy∗ . Then σ(b) = 0, hence 1 = b = ρ(b). Since ρ(1−yy∗) is a projection, it follows that ρ(b) ⩽ ρ(1 − yy∗), hence π(b) ⩽ π 1 − yy∗ . (3.4) Suppose that OL∞(A) λ , and obtain a contradiction. Let F = {b,y,y∗}. Let f and g be as in (3.1) and (3.2). We apply Lemma 2.2 to obtain a finite-dimensional C∗-algebra B, a ucp map ψ : π(A) → B and a unital, self-adjoint map ϕ : B → π(A) such that ϕcb f (λ ), ψϕ = idB, and ϕψ|π(F) = idπ(F). (3.5) By Lemma 2.8, there is a ucp map t : B → B(Hρ) ⊕ B(Hσ ) such that t − ϕcb f (λ ) − 1. (3.6) Let n ∈ N and x ∈ Mn ⊗ B. Since ψϕ = idB, it follows that ϕ(n)(x) ⩾ x. Therefore, t(n) (x) ⩾ ϕ(n) (x) − ϕ(n) (x) − t(n) (x) ⩾ x − f (λ ) − 1 x = 2 − f (λ ) x. Hence t is injective with t−1 cb ⩽ 2 − f (λ ) −1 . (3.7)
- 13. C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 9 Let qρ and qσ denote the orthogonal projections of Hρ ⊕ Hσ onto Hρ ⊕ 0 and 0 ⊕ Hσ respec- tively. By (3.5) and (3.6) we have qσ tψ π(b) ⩽ qσ ϕψ π(b) + f (λ ) − 1 = σ(b) + f (λ ) − 1 = f (λ ) − 1. (3.8) Let p ∈ B be a minimal central projection such that pψ(π(b)) = ψ(π(b)). Then pB ∼ = Mn for some n ∈ N. Using (3.7) and (3.8), we apply Lemma 2.6 with s = qσ t pψ π(b) ⩽ qσ tψ π(b) ⩽ f (λ ) − 1 and r−1 = (t|pB)−1 cb ⩽ 2 − f (λ ) −1 to obtain, (qρt|pB)−1 cb ⩽ 2(2 − f (λ ))2 − 1 2 − f (λ ) −1 . (3.9) Recall that for any finite C∗-algebra C and any contractive x ∈ C, we have 1 − xx∗ = 1 − x∗ x . (3.10) In particular, (3.10) holds for any finite-dimensional C∗-algebra. We will use (3.9) to “isolate” ρ, then the fact that ρ(A) violates (3.10) to arrive at a contradiction: f (λ )−1 ⩽ ϕ−1 cb ⩽ ψ π(b) by (3.5) = pψ π(b) ⩽ p 1 − ψ π(y)π y∗ by (3.4) ⩽ p 1 − ψ π(y) ψ π y∗ (by Lemma 2.5) = p 1 − ψ π y∗ ψ π(y) by (3.10) ⩽ (qρt|pB)−1 cb qρt p 1 − ψ π y∗ ψ π(y) ⩽ (qρt|pB)−1 cb qρt 1 − ψ π y∗ ψ π(y) ⩽ (qρt|pB)−1 cb qρ − qρt ψ π y∗ t ψ π(y) (by Lemma 2.5) ⩽ (qρt|pB)−1 cb qρ − qρϕ ψ π y∗ ϕ ψ π(y) + t − ϕcb 1 + ϕcb = (qρt|pB)−1 cb ρ 1 − y∗ y + t − ϕcb 1 + ϕcb by (3.5) ⩽ 2(2 − f (λ ))2 − 1 2 − f (λ ) −1 f (λ ) − 1 1 + f (λ ) . The last line follows because ρ(y) is an isometry, by (3.5), (3.6) and (3.9). Hence g(f (λ )) 0, a contradiction. 2
- 14. 10 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 In [8] it was asked if there were any nuclear, quasidiagonal C∗-algebras A with OL∞(A) 1. We give some examples of such algebras. Let λ satisfy (3.3). Example 3.2. Let s ∈ B(2) denote the unilateral shift. Then, A = C∗(s ⊕ s∗) is nuclear and quasidiagonal. Applying Theorem 3.1 with ρ : A → C∗(s), σ : A → C∗(s∗) and (xn) = (s ⊕ s∗)n, we have OL∞(A) λ . Before the author obtained Theorem 3.1, Narutaka Ozawa outlined for me an alternate proof that OL∞(C∗(s ⊕ s∗)) 1. The proof was based on the observation that for any finite- dimensional C∗-algebra B and any partial isometry v ∈ B, we have 1 − vv∗ Murray–von Neu- mann equivalent to 1 − v∗v. But, if we let (eij ) denote matrix units for B(2) and T = s ⊕ s∗, then T is a partial isometry and 1 − T ∗T = 0 ⊕ e11 and 1 − T T ∗ = e11 ⊕ 0. So, 1 − T ∗T and 1−T T ∗ are not Murray–von Neumann equivalent in C∗(s ⊕s∗) = B(2)⊕B(2). One can use these facts and arguments similar to Lemmas 2.2 and 2.8 to show that OL∞(C∗(s ⊕ s∗)) 1. Example 3.3. (See [6, Example IX.11.2].) Let D1 and D2 be commuting diagonal operators with joint essential spectrum RP2 , the real projective plane. Let s be as in Example 3.2. Set A = C∗ (s ⊕ D1,0 ⊕ D2). Then, A is easily seen to be an extension of nuclear C∗-algebras and hence is nuclear. As is shown in [6], A is quasidiagonal. Applying Theorem 3.1, with ρ : A → C∗(s), σ : A → C∗(D1,D2) and (xn) = (s ⊕ N1)n, we have OL∞(A) λ . 4. Permanence properties We now investigate a couple permanence properties of OL∞. Let B ⊂ A be nuclear C∗-algebras with OL∞(A) = 1. In general, we do not have OL∞(B) = 1. Indeed let B = C∗(s ⊕ s∗) from Example 3.2. It is easy to see that s ⊕ s∗ is a compact perturbation of a unitary operator u ∈ B(2 ⊕ 2). Let A = C∗(u) + K(2 ⊕ 2). Then A is nuclear and inner quasidiagonal [3, Definition 2.2]. By [3, Theorem 4.5], A is a strong NF algebra, which is a rigid OL∞-space by [2, Theorem 6.1.1]. Hence OL∞(A) = 1, but OL∞(B) 1. In contrast to this situation, if B is an ideal we have the following: Theorem 4.1. Let A be a unital C∗-algebra and J an ideal of A. If OL∞(A) = 1, then OL∞(J) = 1. Proof. Let 0 and E ⊂ J a finite-dimensional subspace. Without loss of generality suppose E has a basis of positive elements x1,...,xn ∈ E with xi = 1 for each i = 1,...,n. Let x1,..., xn ∈ J∗ such that xi, xj = δi,j . Set M = xi. Define δ1(δ) = (1 − δ) − (1 − √ δ )−1 1 − (1 − δ)2 for 0 ⩽ δ 1. Note that δ1(δ) → 1 as δ → 0. Choose δ 0 small enough so 2 √ δ ⩽ /M and (2δ1(δ) − 1)−1 ⩽ 1 + .
- 15. C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 11 By Lemma 2.2 we obtain a finite-dimensional C∗-algebra B = N i=1 Mni , a ucp map ψ :A → B and a unital, self-adjoint map ϕ : B → A with ϕcb ⩽ 1 + δ such that ϕψ|E = idE and ψϕ = idB. We will construct a finite-dimensional subspace E ⊂ F ⊂ J∗∗ such that F is almost com- pletely isometric to a finite-dimensional C∗-algebra and then apply a key theorem from [8] to obtain a subspace E ⊂ F ⊂ J such that F is almost completely isometric to a finite-dimensional C∗-algebra. Since A∗∗ is injective, Lemma 2.8 provides a ucp map t : B → A∗∗ such that t − ϕcb ⩽ δ. Then t is injective and t−1cb ⩽ (1 − δ)−1. Let p ∈ A∗∗ be the central projection such that pA∗∗ = J∗∗. Let t1 : B → J∗∗ be defined by t1(x) = pt(x) and t2 : B → A∗∗ by t2(x) = (1 − p)t(x). Returning to the C∗-algebra B, let q1,...,qN ∈ B be the minimal central projections such that qiB ∼ = Mni . Let I = 1 ⩽ i ⩽ N: sup 1⩽j⩽n qiψ(xj ) ⩽ √ δ . (4.1) Set q = i / ∈I qi and C = qB. We now show that t1 : C → J∗∗ is injective with t1|−1 C cb ⩽ (2δ1(δ) − 1)−1. We first show t1 restricted to each summand of C is almost a complete isometry. To this end, let Ic = {1,...,N}I and j ∈ Ic. Then there is an xi such that qj ψ(xi) √ δ. Since xi ∈ J ∩ E, we have (1 − p)ϕψ(xi) = (1 − p)xi = 0. (4.2) Since ψ is ucp, qj ψ(xi) −1 qj ψ(xi) ∈ qj B ∼ = Mnj (4.3) is norm 1 and positive. Combining (4.2) and (4.3) we obtain t2 qj ψ(xi) −1 qj ψ(xi) ⩽ qj ψ(xi) −1 t2 ψ(xi) = qj ψ(xi) −1 (1 − p)tψ(xi) ⩽ qj ψ(xi) −1 (1 − p)ϕψ(xi) + δ ⩽ √ δ. We apply Lemma 2.6 to t1 : qj B → J∗∗ with s = t2 qj ψ(xi) −1 qj ψ(xi) ⩽ √ δ and r−1 = t|−1 qj B cb ⩽ t|−1 B cb ⩽ (1 − δ)−1 to obtain t1|−1 qj B cb ⩽ δ1(δ)−1 for all j ∈ Ic . (4.4)
- 16. 12 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 Now, let k ∈ N be arbitrary and a = j / ∈I(1k ⊗ qj )a ∈ Mk ⊗ C be positive. Since t1 is com- pletely positive, by (4.4), t (k) 1 (a) ⩾ sup j / ∈I t (k) 1 (1k ⊗ qj )a ⩾ δ1(δ)a. By Lemma 2.4, t1 : C → J∗∗ is injective with t−1 1 cb ⩽ (2δ1(δ) − 1)−1. t1(C) does not necessarily contain x1,...,xn. We fix this with a perturbation. Since pxi = xi = ϕψ(xi) for i = 1,...,n, it follows from (4.1) that xi − t1 ψ(xi)q ⩽ xi − t1ψ(xi) + √ δ = xi − ptψ(xi) + √ δ ⩽ xi − pϕψ(xi) + δ + √ δ = δ + √ δ. Set yi = t1(ψ(xi)q) ∈ J∗∗ for i = 1,...,n. Then, n i=1 xixi − yi ⩽ M(2 √ δ ) ⩽ . By Lemma 2.9 there is a complete isomorphism w : J∗∗ → J∗∗ such that w(yi) = xi for i = 1,...,n and wcbw−1cb ⩽ (1 + )/(1 − ). Let F = wt1(C) ⊂ J∗∗. Then E ⊂ F and dcb( F,C) ⩽ (1 + ) (1 − ) 2δ1(δ) − 1 −1 (1 + )2 1 − . By [8, Theorem 4.3] there is a subspace F ⊂ J such that E ⊂ F and dcb(F,C) (1 + )2(1 − )−1. Since 0 was arbitrary, it follows that OL∞(J) = 1. 2 Remark 4.2. It is not known if Theorem 4.1 holds in general, i.e. if J is an ideal of A do we always have OL∞(J) ⩽ OL∞(A)? Remark 4.3. Blackadar and Kirchberg have shown [2, Proposition 6.1.7] that every hereditary subalgebra of a rigid OL∞ space is also a rigid OL∞ space. It is not known if Theorem 4.1 can be extended to include hereditary sub C∗-algebras. Finally, we need the following Proposition for Section 5. For C∗-algebras A and B, let A B denote the algebraic tensor product of A and B. Proposition 4.4. Let A1 and A2 be nuclear C∗-algebras. Then OL∞(A1 ⊗ A2) ⩽ OL∞(A1)OL∞(A2).
- 17. C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 13 Proof. Let E ⊂ A1 A2 be a finite-dimensional subspace and 0. For i = 1,2 choose finite- dimensional subspaces Fi ⊂ Ai, finite-dimensional C∗-algebras Bi and linear isomorphisms ϕi :Fi → Bi, such that ϕicb ϕ−1 i cb ⩽ OL∞(Ai) + and E ⊂ F1 F2. Let ⊗min denote the minimal operator space tensor product. Recall that for C∗-algebras the minimal operator space tensor product coincides with the minimal C∗-tensor product (see [12, p. 228]). Furthermore by [12, 2.1.3], ϕ1 ⊗ ϕ2 : F1 ⊗min F2 → B1 ⊗ B2cb ⩽ ϕ1cbϕ2cb. We have a similar inequality for (ϕ1 ⊗ ϕ2)−1 = ϕ−1 1 ⊗ ϕ−1 2 . Since A1 A2 is dense in A1 ⊗ A2, it follows that OL∞(A1 ⊗ A2) ⩽ inf 0 OL∞(A1) + OL∞(A2) + = OL∞(A1)OL∞(A2). 2 5. Irreducible representations and OL∞ This section contains the main theorem (Theorem 5.4). We first recall the necessary definitions and prove some preliminary lemmas. Definition 5.1. Let A be a unital C∗-algebra. Recall that x ∈ A is an isometry if x∗x = 1. An isometry is called proper, if xx∗ = 1. A is called finite if it contains no proper isometries. A is called stably finite if Mn ⊗ A is finite for every n ∈ N. We will call a representation π of A finite (resp. stably finite) if A/ker(π) is finite (resp. stably finite). Lemma 5.2. Let H be a separable Hilbert space and x ∈ B(H) be a proper isometry. Then there is a unitary u ∈ B(H) such that (ux)n(ux)∗n → 0 strongly. Proof. It is well known (see [6, Theorem V.2.1]) that there is a closed subspace K ⊂ H such that relative to the decomposition H = K ⊕ K⊥, we have x = s ⊕ w where s ∈ B(K) is unitarily equivalent to sα, the unilateral shift of order α (for some α = 1,2,...,∞), and w is a unitary in B(K⊥). In particular sns∗n → 0 strongly in B(K). Without loss of generality, assume that w = idK⊥ . Suppose first that K⊥ is infinite-dimensional. Since x|K is a proper isometry, K is also infinite-dimensional. Since H is separable, K ∼ = K⊥. Under this identification and relative to the decomposition H = K ⊕ K, let u = 0 1 1 0 ∈ B(H). Then for n ∈ N we have, (ux)2n (ux)∗2n = sns∗n 0 0 sns∗n
- 18. 14 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 and (ux)2n+1 (ux)∗(2n+1) = sns∗n 0 0 s(n+1)s∗(n+1) . Hence, (ux)n(ux)∗n → 0 strongly. Suppose now that dim(K⊥) = n ∞. Let {f1,...,fn} be an orthonormal basis for K⊥. Since s is unitarily equivalent to a shift, let e1,...,en ∈ K be an orthonormal set such that sei = ei+1 for i = 1,...,n − 1 and e1 ⊥ x(H). Define u ∈ B(H) by u(ei) = fi and u(fi) = ei for i = 1,...,n and u(η) = η for η ⊥ span{e1,...,en,f1,...,fn}. Then u is unitary and (ux)2n (H) ⊥ span{e1,...,en,f1,...,fn}. Hence for every k ⩾ 2n we have (ux)2n+k = xk(ux)2n. Therefore, (ux)2n+k (ux)∗(2n+k) ⩽ sk ⊕ 0K⊥ sk ⊕ 0K⊥ ∗ → 0 strongly. 2 We recall the following definitions (see [10, Section 4.1]). Let A be a C∗-algebra. An ideal J of A is called primitive if J is the kernel of some (non- zero) irreducible representation of A. Let Prim(A) denote the set of all primitive ideals of A. For a subset X ⊂ Prim(A), and an ideal J of A let ker(X) = I∈X I and hull(J) = I ∈ Prim(A): J ⊂ I . Then Prim(A) is a topological space with closure operation X → hull(ker(X)) (see [10, Theo- rem 4.1.3]). The following is an easy consequence of [10, Theorem 4.1.3]. Lemma 5.3. Let A be a C∗-algebra and X ⊂ Prim(A). Then X is dense if and only if ker(X) = {0}. Theorem 5.4. Let A be a separable unital C∗-algebra with OL∞(A) λ , where λ satis- fies (3.3). Then A has a separating family of irreducible, stably finite representations. Proof. We first show that A has a separating family of irreducible, finite representations. We assume that A does not have a separating family of irreducible, finite representations and prove that OL∞(A) λ . Let Y = y ∈ A: ∃J ∈ Prim(A) (y + J ∈ A/J is a proper isometry) . Then Y is not empty. For each y ∈ Y, let O(y) = J ∈ Prim(A): 1 − y∗ y + J 1/4 and 1 − yy∗ + J 3/4 , (5.1)
- 19. C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 15 CO(y) = Prim(A) hull ker O(y) . We will now prove the following statement: (∃y ∈ Y) CO(y) is not dense in Prim(A) . (5.2) (If Prim(A) is Hausdorff, then (5.2) is immediate by [10, Proposition 4.4.5]. But Prim(A) is not Hausdorff in general.) Since A is separable, let (yn) ⊂ Y be a dense sequence. Suppose that (5.2) does not hold. Then CO(yn) is a dense, open subset of Prim(A) for each n ∈ N. Since Prim(A) is a Baire space, (see [10, Theorem 4.3.5]) the following set is dense in Prim(A): X = ∞ n=1 CO(yn). If there is a J ∈ X such that A/J is not finite, then there is a y ∈ Y such that y + J is a proper isometry. Then there is an n ∈ N such that yny∗ n − yy∗ + y∗ nyn − y∗ y 1/8. But this implies that J ∈ O(yn) ∩ X ⊂ hull ker O(yn) ∩ X = ∅. Hence for every J ∈ X, A/J is finite. Since X is dense, ker(X) = {0} by Lemma 5.3. Then A has a separating family of irreducible finite representations, a contradiction. This completes the proof of (5.2). We now build representations ρ and σ that satisfy Theorem 3.1. Let y ∈ Y satisfy (5.2). For each J ∈ CO(y) let σJ be an irreducible representation of A such that ker(σJ ) = J . Let σ = J∈CO(y) σJ . Since CO(y) is not dense, we have ker(σ) = J∈CO(y) J = ker CO(y) = {0}. (5.3) Let {Ji}i∈I ⊂ O(y) be an at most countable subset such that ker {Ji}i∈I = ker O(y) . (5.4) For i ∈ I, let ρi be an irreducible representation of A such that ker(ρi) = Ji. Let ρ = i∈I ρi. By (5.3) and (5.4) we have ker(ρ ⊕ σ) = ker O(y) ∩ J∈CO(y) J = J∈Prim(A) J = {0}. (5.5)
- 20. 16 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 By definition (5.1), for every i ∈ I, we have 1 − ρi(y∗y) 1/4. Hence ρi(y) is left invertible and ρi(y∗y) is invertible. We note that ρi(y) is not right invertible. Indeed, if ρi(y) is right invertible, then there is a unitary u ∈ ρi(A) such that ρi(y) = u|ρi(y)|. Then by (5.1) we have 3/4 1 − ρi yy∗ = 1 − uρi y∗ y u∗ = 1 − ρi y∗ y 1/4, a contradiction. For each i ∈ I, let zi = ρi(y) ρi y∗ y −1/2 . Then z∗ i zi = 1, but ziz∗ i = 1 because ρi(y) is not right invertible. Hence, zi ∈ ρi(A) is a proper isometry for each i ∈ I. Define the continuous function f : R+ → R+ by f (t) = 8 3 √ 3 t if 0 ⩽ t ⩽ 3/4, t−1/2 if t 3/4. Let x = yf (y∗y) ∈ A. Since sp(ρi(y∗y)) ⊂ [3/4,1], it follows that ρi( x ) = zi for each i ∈ I. Let x ∈ A be norm 1 such that ρ(x) = ρ( x ) (such a lifting is always possible, see [16, Remark 8.6]). Let Hi denote the Hilbert space associated with ρi. For each i ∈ I, Lemma 5.2 provides a unitary ui ∈ B(Hi) such that (uizi)n (uizi)∗n → 0 strongly in B(Hi), as n → ∞. (5.6) Since each ρi has a different kernel, they are mutually inequivalent. So, by [10, Theorem 3.8.11] ρ(A) = i∈I ρi(A) = i∈I B(Hi). Set u = i∈I ui. Since ρi(x) = zi, by (5.6) we have uρ(x) n uρ(x) ∗n → 0 strongly in i∈I B(Hi). By Kaplansky’s density theorem (see [14, Theorem II.4.11]) there is a sequence (uk) of unitaries from ρ(A) such that uk → u in the strong* topology. From this we obtain sequences (kr) and (nr) such that ukr ρ(x) nr ukr ρ(x) ∗nr → 0 strongly as r → ∞. (5.7) For each r ∈ N let xr ∈ A be norm 1 such that ρ(xr) = ukr . By (5.3) and (5.5) we apply Theo- rem 3.1 with the sequence (xrx)∞ r=1 and deduce that OL∞(A) λ . We now return to the general case. Suppose that OL∞(A) λ . Let H be a separable, infinite-dimensional Hilbert space. Let K denote the compact oper- ators on H and K1 be the unitization of K. Since K1 is an AF algebra, OL∞(K1) = 1. By Proposition 4.4, OL∞(A ⊗ K1) ⩽ OL∞(A) λ .
- 21. C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 17 By the above proof there is a subset X ⊂ Prim(A ⊗ K1) with ker(X) = {0} and (A ⊗ K1)/J finite for each J ∈ X. By [1, IV.3.4.23], Prim A ⊗ K1 = J ⊗ K1 + A ⊗ I: J ∈ Prim(A), I = {0}, K . So, without loss of generality we may assume X = {Ji ⊗ K1}i∈I with Ji ∈ Prim(A). Since K1 is exact, (A ⊗ K1)/(Ji ⊗ K1) = (A/Ji) ⊗ K1, so A/Ji is stably finite. Furthermore, by the exactness of K1, we have {0} = i∈I Ji ⊗ K1 = i∈I Ji ⊗ K1 . So, ker({Ji}i∈I ) = {0}. 2 We are now in a position to give a new class of examples of nuclear, quasidiagonal C∗- algebras A with OL∞(A) 1. Example 5.5. Let A be a unital nuclear C∗-algebra without a separating family of irreducible stably finite representations (in particular any non-finite nuclear, C∗-algebra). Let C(A)1 = (C0(0,1] ⊗ A)1 be the unitization of the cone of A. Since A is nuclear, so is C(A)1. By [15, Proposition 3] C(A)1 is quasidiagonal. For t ∈ (0,1], let It = {f ∈ C0(0,1]: f (t) = 0}. By [1, IV.3.4.23] every non-essential primitive ideal of C(A)1 is of the form It ⊗ A + C0(0,1] ⊗ J for some J ∈ Prim(A) and 0 t ⩽ 1. Furthermore, by [1, IV.3.4.22], C0(0,1] ⊗ A / It ⊗ A + C0(0,1] ⊗ J ∼ = A/J. From this we deduce that C(A)1 cannot have a separating family of irreducible, stably finite representations, hence OL∞(C(A)1) λ by Theorem 5.4. 6. Questions and remarks Recall from the Introduction: Question 6.1. (See [8, Question 6.1].) If OL∞(A) = 1, is A a rigid OL∞ space? Blackadar and Kirchberg showed [3, Theorem 4.5] that a C∗-algebra A is nuclear and inner quasidiagonal if and only if A is a strong NF algebra (see [2, Definition 5.2.1]). In [8] it was shown that A is a strong NF algebra if and only if A is a rigid OL∞ space. Furthermore, by [3, Proposition 2.4] any C∗-algebra with a separating family of irreducible quasidiagonal represen- tations is inner quasidiagonal. Therefore if there is a C∗-algebra A with OL∞(A) = 1, but which is not a rigid OL∞ space, then A cannot have a separating family of irreducible, quasidiagonal representations, but A must
- 22. 18 C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 have a separating family of irreducible stably finite representations by Theorem 5.4. Let A ⊂ B(H) from Example 3.3. Then A + K(H) is stably finite and prime, hence has a faithful stably finite representation. On the other hand by [4], the unique irreducible representation of A+K(H) is not quasidiagonal. Hence, A + K(H) is a possible counterexample to Question 6.1. Finally, recall the question raised by Blackadar and Kirchberg: Question 6.2. (See [2, Question 7.4].) Is every nuclear stably finite C∗-algebra quasidiagonal? There are some interesting relationships between Question 6.2 and OL∞ structure. Proposition 6.3. Let A be either simple or both prime and antiliminal. If 1 OL∞(A) 1 + √ 5 2 1/2 then A is (nuclear) stably finite, but not quasidiagonal. Proof. This follows from [3, Corollary 2.6] and [8, Theorem 3.4]. 2 In light of Theorem 5.4, we have the following similar relationship: Proposition 6.4. Let A be a C∗-algebra such that every primitive quotient is antiliminal. If 1 OL∞(A) 1.005 then some quotient of A is (nuclear) stably finite, but not quasidiagonal. Proof. This follows from [3, Corollary 2.6] and Theorem 5.4. 2 Acknowledgments A portion of the work for this paper was completed while the author took part in the Thematic Program on Operator Algebras at the Fields Institute in Toronto, ON in the Fall of 2007. I would like to thank Narutaka Ozawa for a helpful discussion about this work and my advisor Zhong-Jin Ruan for all his support. References [1] B. Blackadar, Operator algebras, in: Theory of C∗-Algebras and von Neumann Algebras, Operator Algebras and Non-Commutative Geometry, III, in: Encyclopaedia Math. Sci., vol. 122, Springer-Verlag, Berlin, 2006. [2] Bruce Blackadar, Eberhard Kirchberg, Generalized inductive limits of finite-dimensional C*-algebras, Math. Ann. 307 (3) (1997) 343–380. [3] Bruce Blackadar, Eberhard Kirchberg, Inner quasidiagonality and strong NF algebras, Pacific J. Math. 198 (2) (2001) 307–329. [4] L.G. Brown, The universal coefficient theorem for Ext and quasidiagonality, in: Operator Algebras and Group Representations, vol. I, Neptun, 1980, in: Math. Stud. Monogr. Ser., vol. 17, Pitman, Boston, MA, 1984, pp. 60–64. [5] Nathanial P. Brown, On quasidiagonal C∗-algebras, in: Operator Algebras and Applications, in: Adv. Stud. Pure Math., vol. 38, Math. Soc. Japan, Tokyo, 2004, pp. 19–64.
- 23. C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19 19 [6] Kenneth R. Davidson, C∗-Algebras by Example, Fields Inst. Monogr., vol. 6, American Mathematical Society, Providence, RI, 1996. [7] M. Junge, N.J. Nielsen, Zhong-Jin Ruan, Q. Xu, COLp spaces—the local structure of non-commutative Lp spaces, Adv. Math. 187 (2) (2004) 257–319. [8] Marius Junge, Narutaka Ozawa, Zhong-Jin Ruan, On OL∞ structures of nuclear C∗-algebras, Math. Ann. 325 (3) (2003) 449–483. [9] Vern Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Stud. Adv. Math., vol. 78, Cambridge University Press, Cambridge, 2002. [10] Gert K. Pedersen, C∗-Algebras and Their Automorphism Groups, London Math. Soc. Monogr. Ser., vol. 14, Aca- demic Press Inc., London, 1979. [11] Gilles Pisier, The operator Hilbert space OH, complex interpolation and tensor norms, Mem. Amer. Math. Soc. 122 (585) (1996). [12] Gilles Pisier, Introduction to Operator Space Theory, London Math. Soc. Lecture Note Ser., vol. 294, Cambridge University Press, Cambridge, 2003. [13] R.R. Smith, Completely bounded maps between C∗-algebras, J. London Math. Soc. (2) 27 (1) (1983) 157–166. [14] M. Takesaki, Theory of Operator Algebras, I, Encyclopaedia Math. Sci., vol. 124, Springer-Verlag, Berlin, 2002. Reprint of the first edition, 1979, Operator Algebras and Non-Commutative Geometry, vol. 5. [15] Dan Voiculescu, A note on quasi-diagonal C∗-algebras and homotopy, Duke Math. J. 62 (2) (1991) 267–271. [16] Simon Wassermann, Exact C∗-Algebras and Related Topics, Lect. Notes Ser., vol. 19, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994.
- 24. Journal of Functional Analysis 258 (2010) 20–49 www.elsevier.com/locate/jfa Groupoid normalizers of tensor products Junsheng Fang a , Roger R. Smith a,∗,1 , Stuart A. White b , Alan D. Wiggins c a Department of Mathematics, Texas AM University, College Station, TX 77843, USA b Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK c Department of Mathematics and Statistics, University of Michigan–Dearborn, Dearborn, MI 48128, USA Received 15 September 2008; accepted 7 October 2009 Communicated by N. Kalton Abstract We consider an inclusion B ⊆ M of finite von Neumann algebras satisfying B ∩ M ⊆ B. A partial isometry v ∈ M is called a groupoid normalizer if vBv∗,v∗Bv ⊆ B. Given two such inclusions Bi ⊆ Mi, i = 1,2, we find approximations to the groupoid normalizers of B1 ⊗ B2 in M1 ⊗ M2, from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis B i ∩Mi ⊆ Bi, i = 1,2. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries v ∈ M satisfying vBv∗ ⊆ B and v∗v,vv∗ ∈ B. © 2009 Elsevier Inc. All rights reserved. Keywords: Groupoid normalizer; Tensor product; von Neumann algebra; Finite factor 1. Introduction The focus of this paper is an inclusion B ⊆ M of finite von Neumann algebras. Such inclusions have a rich diverse history, first being studied by Dixmier [3] in the context of maximal abelian subalgebras (masas) of II1 factors. These inclusions provided the basic building blocks for the * Corresponding author. E-mail addresses: jfang@math.tamu.edu (J. Fang), rsmith@math.tamu.edu (R.R. Smith), s.white@maths.gla.ac.uk (S.A. White), adwiggin@umd.umich.edu (A.D. Wiggins). 1 Partially supported by a grant from the National Science Foundation. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.005
- 25. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 21 theory of subfactors developed by Jones in [9] and today they are a key component in the study of structural properties of II1 factors using the deformation-rigidity techniques introduced by Popa in [14]. In [3], Dixmier introduced a classification of masas in II1 factors using normalizers, defining NM(B) = {u a unitary in M: uBu∗ = B}. A masa B ⊂ M is Cartan or regular if these normal- izers generate M and singular if NM(B) ⊂ B. Feldman and Moore demonstrated the importance of Cartan masas, and hence normalizers, in the study of II1 factors, showing that inclusions of Cartan masas arise from measurable equivalence relations and that, up to orbit equivalence, these relations determine the resulting inclusion [7,8]. Given two inclusions Bi ⊂ Mi of masas in II1 factors, it is immediate that an elementary tensor u1 ⊗u2 of unitaries ui ∈ Mi normalizes the tensor product inclusion B = B1 ⊗B2 ⊂ M = M1 ⊗ M2 if and only if each ui normalizes Bi. As a simple consequence, the tensor product of Cartan masas is again Cartan. More generally, the operation of passing to the von Neumann algebra generated by the normalizers was shown to commute with the tensor product operation for masas inside II1 factors, in the sense that the equality NM1 (B1) ⊗ NM2 (B2) = NM1⊗M2 (B1 ⊗ B2) (1.1) holds. This was proved when both masas are singular in [19] and the general case was established by Chifan in [1]. Since the containment from left to right in (1.1) is immediate, the problem in both cases is to eliminate the possibility that some unexpected unitary in the tensor product normalizes B1 ⊗ B2. This difficulty was overcome in [19] and [1] by employing techniques of Popa [15] to analyse the basic construction algebra M,eB of Jones [9]. Beyond the masa setting, (1.1) holds when each Bi satisfies B i ∩ Mi = C1, the defining property of irreducible subfactors. When each Bi has finite Jones index in Mi, the identity (1.1) can be deduced from results of [13]. The infinite index case was established in [20], where every normalizing unitary of such a tensor product of irreducible subfactors was shown to be of the form w(v1 ⊗v2), where w is a unitary in B1 ⊗ B2 and each vi ∈ NMi (Bi). Some other situations where (1.1) holds are discussed in [6]. For general inclusions Bi ⊆ Mi of finite von Neumann algebras, the commutation iden- tity (1.1) can fail. Indeed, taking each Mi to be a copy of the 3 × 3 matrices and each Bi ∼ = C ⊕ M2(C), one obtains inclusions with NMi (Bi) ⊂ Bi, yet there are non-trivial normaliz- ers of B1 ⊗B2 inside M1 ⊗M2. This is due to the presence of partial isometries v in Mi Bi with vBiv∗ ⊆ Bi and v∗Biv ⊆ Bi, as the non-trivial unitary normalizers of B1 ⊗ B2 can all be writ- ten in the form j xj (v1,j ⊗ v2,j ), where xj lie in B1 ⊗ B2 and the vi,j are partial isometries with vi,j Biv∗ i,j ⊆ B and v∗ i,j Bivi,j ⊆ Bi. Defining the groupoid normalizers of a unital inclu- sion B ⊂ M to be the set GNM(B) = {v a partial isometry in M: vBv∗ ⊆ B,v∗Bv ⊆ B}, the example discussed above satisfies the commutation identity GNM1 (B1) ⊗ GNM2 (B2) = GNM1⊗M2 (B1 ⊗ B2) . (1.2) In this paper we examine groupoid normalizers of tensor product algebras, establishing (Corol- lary 5.6) the identity (1.2) whenever Bi ⊆ Mi are inclusions of finite von Neumann algebras with separable preduals satisfying B i ∩ Mi ⊆ Bi for each i. In [4] Dye shows that every groupoid normalizer v of a masa B in M is of the form v = ue for some projection e = v∗v ∈ B and some unitary normalizer u of B in M, see also [18, Lemma 6.2.3]. The same result holds by a direct computation when B is an irreducible subfactor of M, so that in these two cases
- 26. 22 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 NM(B) = GNM(B) and (1.2) directly generalizes (1.1) established in [1] and [20] respec- tively. The following example shows why the hypothesis B i ∩ Mi ⊆ Bi (which is satisfied by both masas and irreducible subfactors) is necessary in this result. Example 1.1. Consider the subalgebra B = α 0 0 0 α 0 0 0 β : α,β ∈ C ⊆ M3, and note that B ∩ M3 strictly contains B. A direct computation shows that GN(B) = M2 ⊕ C, and so GNM3 (B) ⊗ GNM3 (B) ∼ = M4 ⊕ M2 ⊕ M2 ⊕ C. However, B ⊗ B is isomorphic to CI4 ⊕ CI2 ⊕ CI2 ⊕ C inside M9, and GNM3⊗M3 (B ⊗ B) is M4 ⊕ M4 ⊕ C. A new feature of [20] was the notion of one-sided normalizers of an irreducible inclusion B ⊂ M of II1 factors, namely those unitaries u ∈ M with uBu∗ B. These cannot arise for finite index inclusions by index considerations, or in the case when B ⊂ M is a masa. To establish (1.1) for irreducible subfactors, it was necessary to first establish the general form of a one-sided normalizer of a tensor product of irreducible subfactors and then deduce the normalizer result from this. The same procedure is necessary here, so we introduce the notion of an intertwiner to study groupoid normalizers in a one-sided situation. Definition 1.2. Given an inclusion B ⊆ M of von Neumann algebras satisfying B ∩ M ⊆ B, define the collection GN (1) M (B) of intertwiners of B in M by GN (1) M (B) = v a partial isometry in M: vBv∗ ⊆ B,v∗ v ∈ B . We will write GN(1) (B) for GN (1) M (B) when there is no confusion about the underlying algebra M. We use the superscript (1) to indicate that our intertwiners are one-sided, namely that although vBv∗ ⊆ B, we are not guaranteed to have a containment v∗Bv ⊆ B. Note that v ∈ GNM(B) if, and only if, both v and v∗ lie in GN (1) M (B). Note too that while the groupoid normalizers form a groupoid, the intertwiners do not. Finally, the terminology intertwiner comes from the fact that, under the hypothesis B ∩ M ⊆ B, these are exactly the partial isometries that witness the embeddability of a corner of B into itself inside M in the sense of Popa’s intertwining procedure for subalgebras from [14,15]. We obtain a similar commutation result to (1.2) for intertwiners. In fact our main theorem, stated below, obtains more as it gives approximate forms for intertwiners and groupoid normal- izers of tensor products. Theorem 1.3. Let Bi ⊂ Mi be inclusions of finite von Neumann algebras with separable preduals and with fixed faithful normal traces τi on Mi. Moreover, suppose that B i ∩Mi ⊆ Bi for i = 1,2. For v ∈ GN(1) M1⊗M2 (B1 ⊗ B2) and ε 0, there exist k ∈ N and operators x1,...,xk ∈ B1 ⊗ B2, intertwiners w1,1,...,w1,k of B1 in M1 and intertwiners w2,1,...,w2,k of B2 in M2 such that
- 27. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 23 v − k j=1 xj (w1,j ⊗ w2,j ) 2 ε, (1.3) where the · 2-norm arises from the trace τ1 ⊗ τ2 on M1 ⊗ M2. If in addition v is a groupoid normalizer, then each wi,j can be taken to be a groupoid normalizer rather than just an inter- twiner. The intertwiner form of Theorem 1.3 is established as Theorem 4.7 and additional analysis in Section 5 enables us to deduce the groupoid normalizer form of Theorem 1.3 as Theorem 5.5. For the remainder of the introduction we give a summary of the main steps used to establish these results and where they can be found in the paper. Given inclusions Bi ⊆ Mi of finite von Neumann algebras with B i ∩ Mi ⊆ Bi, write B ⊂ M for the tensor product inclusion B1 ⊗B2 ⊆ M1 ⊗M2. Let v ∈ GN (1) M (B). Then the element v∗eBv is a projection in the basic construction algebra M,eB , the properties of which are recalled in Section 2. Section 3 discusses the properties of these projections in the basic construction arising from intertwiners. In particular, we show that the projection v∗eBv is central in the cut-down (B ∩ M,eB )v∗v (Lemma 3.2) and construct an explicit projection Pv ∈ Z(B ∩ M,eB ) with Pvv∗v = v∗eBv. We need to construct this projection explicitly rather than appeal to general theory, as its properties (established in Lemma 3.8) are crucial subsequently. Since the basic construction factorizes as a tensor product M,eB ∼ = M1,eB1 ⊗ M2,eB2 , Tomita’s commutation theorem gives Z B ∩ M,eB ∼ = Z B 1 ∩ M1,eB1 ⊗ Z B 2 ∩ M2,eB2 . (1.4) For each i = 1,2, let Qi denote the supremum of all projections in Z(B i ∩ Mi,eBi ) of the form j w∗ i,j eBi wi,j , where the wi,j lie in GN (1) Mi (Bi) and satisfy wi,j w∗ i,k = 0 when j = k. If we can show that Pv ⩽ Q1 ⊗ Q2, (1.5) then it will follow that we can approximate Pv in L2( M,eB ) by projections of the form j (w1,j ⊗ w2,j )∗eB(w1,j ⊗ w2,j ) for intertwiners wi,j ∈ GN (1) Mi (Bi). To do this, we use the fact that projections in the tensor product (1.4) of abelian von Neumann algebras can be approxi- mated by sums of elementary tensors of projections, and so it is crucial that the original projection v∗eBv be central in (B ∩ M,eB )v∗v, for which the hypothesis B ∩ M ⊆ B is necessary. Fi- nally, we push the approximation for Pv down to M and obtain the required approximation for v in M (see Theorem 4.7). Most of Section 4 is taken up with establishing (1.5). We give a technical result (Theorem 4.1), which in particular characterizes when a projection in the basic construction arises from an in- tertwiner. By applying Theorem 4.1 to Pv and the inclusion Z B 1 ∩ M1,eB1 ⊗ B2 ⊆ Z B 1 ∩ M1,eB1 ⊗ M2, regarded as a direct integral of inclusions of finite von Neumann algebras, we are able to establish Pv ⩽ 1 ⊗ Q2 in Lemma 4.6 and so (1.5) follows by symmetry. It should be noted that the intro- duction of the projections Qi is essential in order to make use of measure theory, particularly the
- 28. 24 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 uniqueness of product measures on σ-finite spaces [17, p. 312]. The canonical trace on the basic construction need not be a semifinite weight on Z(B i ∩ Mi,eBi ) but does have this property on the compression Z(B i ∩ Mi,eBi )Qi where it can be treated as a measure (see Lemma 2.6 and the discussion preceding Definition 4.4). The remaining difficulty is to check that the pro- jection Pv satisfies the hypotheses of Theorem 4.1, for which we require certain order properties of the pull down map on the basic construction. These are described in the next section, in which we also set out our notation, review the properties of the basic construction, and establish some technical lemmas. Finally, the paper ends with Section 5, which handles the additional details required to deduce the groupoid normalizer result (Theorem 5.5) from our earlier work. 2. Notation and preliminaries Throughout the paper, all von Neumann algebras are assumed to have separable preduals. The basic object of study in this paper is an inclusion B ⊆ M of finite von Neumann algebras, where M is equipped with a faithful normal trace τ satisfying τ(1) = 1. We always assume that M is standardly represented on the Hilbert space L2(M,τ), or simply L2(M). The letter ξ is reserved for the image of 1 ∈ M in this Hilbert space, and J will denote the isometric conjugate linear operator on L2(M) defined on Mξ by J(xξ) = x∗ξ, x ∈ M, and extended by continuity to L2(M) from this dense subspace. Then L2(B) is a closed subspace of L2(M), and eB denotes the projection of L2(M) onto L2(B), called the Jones projection. The von Neumann algebra generated by M and eB is called the basic construction and is denoted by M,eB [2,9]. Let EB denote the unique trace preserving conditional expectation of M onto B. In the next proposition we collect together standard properties of eB,EB and M,eB from [9,13,10,18]. Proposition 2.1. (i) eB(xξ) = EB(x)ξ, x ∈ M. (ii) eBxeB = EB(x)eB = eBEB(x), x ∈ M. (iii) M ∩ {eB} = B. (iv) M,eB = JBJ , Z( M,eB ) = JZ(B)J . (v) eB has central support 1 in M,eB . (vi) Span{xeBy: x,y ∈ M} generates a ∗-strongly dense subalgebra, denoted MeBM, of M,eB . (vii) x → eBx and x → xeB are injective maps for x ∈ M. (viii) MeB and eBM are ∗-strongly dense in M,eB eB and eB M,eB respectively. (ix) eB M,eB eB = BeB = eBB. (x) (MeBM) M,eB (MeBM) ⊆ MeBM. (xi) There is a unique faithful normal semifinite trace Tr on M,eB satisfying Tr(xeBy) = τ(xy), x,y ∈ M. (2.1) This trace is given by the formula Tr(t) = ∞ i=1 tJv∗ i ξ,Jv∗ i ξ , t ∈ M,eB + , (2.2) where the vi’s are partial isometries in M,eB satisfying ∞ i=1 v∗ i eBvi = 1.
- 29. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 25 (xii) The algebra MeBM is · 2,Tr-dense in L2( M,eB ,Tr) and · 1,Tr-dense in L1( M,eB ,Tr). (xiii) Given inclusions Bi ⊂ Mi of finite von Neumann algebras for i = 1,2, the basic construc- tion M1 ⊗ M2,eB1⊗B2 is isomorphic to M1,eB1 ⊗ M2,eB2 . Under this isomorphism, the canonical trace Tr on M1 ⊗M2,eB1⊗B2 is given by Tr1 ⊗Tr2, where Tri is the canon- ical trace on Mi,eBi . (xiv) There is a well-defined map Ψ : MeBM → M, given by Ψ (xeBy) = xy, x,y ∈ M. (2.3) This is the pull down map of [13], where it was shown to extend to a contraction from L1( M,eB ,Tr) to L1(M,τ). Using part (xii) of the previous proposition, the equation Tr (xeBy)z = τ (xy)z = τ Ψ (xeBy)z , x,y,z ∈ M, (2.4) shows that Ψ is the pre-adjoint of the identity embedding M → M,eB and is, in particular, positive. The basic properties of Ψ are set out in [13], but we will need more detailed informa- tion on this map than is currently available in the literature. We devote much of this section to obtaining further properties of Ψ , the main objective being to apply them in Lemma 4.5. In the next three lemmas, the inclusion B ⊂ M is always of arbitrary finite von Neumann algebras with a fixed faithful normalized normal trace τ on M, inducing the trace Tr on M,eB . Lemma 2.2. Let x ∈ L1( M,eB )+ ∩ M,eB . If Ψ (x) ∈ L1(M) ∩ M, then Ψ (x) ⩾ x. Proof. It suffices to show that Ψ (x)yξ,yξ ⩾ xyξ,yξ , y ∈ M. (2.5) The maximality argument, preceding [18, Lemma 4.3.4], to establish part (xi) of Proposition 2.1 can be easily modified to incorporate the requirement that v1 = 1. Thus there are vectors ξi = Jv∗ i ξ ∈ L2(M) so that (2.2) becomes Tr(t) = ∞ i=1 tξi,ξi , t ∈ M,eB + , (2.6) where ξ1 = ξ. Now, for y ∈ M, we may use the M-modularity of Ψ to write Ψ (x)yξ,yξ = Ψ y∗ xy ξ,ξ = τ Ψ y∗ xy = Tr y∗ xy . (2.7) It follows from (2.6) and (2.7) that Ψ (x)yξ,yξ = xyξ,yξ + ∞ i=2 xyξi,yξi ⩾ xyξ,yξ , y ∈ M, (2.8) establishing that Ψ (x) ⩾ x. 2
- 30. 26 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 We now extend this result to tensor products. Let N be a semifinite von Neumann algebra with a specified faithful normal semifinite trace TR. In [5], Effros and Ruan identified the pre- dual of a tensor product of von Neumann algebras X and Y by (X ⊗ Y)∗ = X∗ ⊗op Y∗, the operator space projective tensor product of the preduals. In the presence of traces, this identi- fies L1(X ⊗ Y) with L1(X) ⊗op L1(Y), so I ⊗ Ψ is well defined, positive, and bounded from L1(N ⊗ M,eB ,TR ⊗ Tr) to L1(N ⊗ M,TR ⊗ τ), being the pre-adjoint of the identity em- bedding N ⊗ M → N ⊗ M,eB . Following [21, Chapter IX], we will always assume that N is faithfully represented on L2(N,TR), for which span{y ∈ N: TR(y∗y) ∞} is a dense subspace. Lemma 2.3. Let x ∈ L1(N ⊗ M,eB )+ ∩ (N ⊗ M,eB ). If (I ⊗ Ψ )(x) ∈ L1(N ⊗ M) ∩ (N ⊗ M), then (I ⊗ Ψ )(x) ⩾ x. Proof. Suppose that the result is not true. Then we may find a finite projection p ∈ N, elements yi ∈ pNp and zi ∈ M, 1 ⩽ i ⩽ k, so that x − (I ⊗ Ψ )(x) k i=1 yi ⊗ ziξ, k i=1 yi ⊗ ziξ 0, (2.9) since such sums k i=1 yi ⊗ ziξ are dense in L2(N,TR) ⊗2 L2(M,τ). Then the inequality (I ⊗ Ψ ) (p ⊗ 1)x(p ⊗ 1) ⩾ (p ⊗ 1)x(p ⊗ 1) (2.10) fails. The element on the left of (2.10) is (p ⊗ 1)((I ⊗ Ψ )(x))(p ⊗ 1), and so is bounded by hy- pothesis. The restriction of I ⊗ Ψ to L1(pNp ⊗ M,eB ) is the pull down map for the inclusion pNp ⊗ B ⊆ pNp ⊗ M of finite von Neumann algebras with basic construction pNp ⊗ M,eB . The failure of (2.10) then contradicts Lemma 2.2 applied to this inclusion, establishing that (1 ⊗ Ψ )(x) ⩾ x. 2 The next lemma completes our investigation of the order properties of pull down maps. Lemma 2.4. If x ∈ L1(N ⊗ M,eB )+ is unbounded, then so also is (1 ⊗ Ψ )(x). Proof. Suppose that (1 ⊗ Ψ )(x) is bounded. Following [21, Section IX.2], we may regard x as a self-adjoint positive densely defined operator on L2(N ⊗ M,eB ). For n ⩾ 1, let pn ∈ N be the spectral projection of x for the interval [0,n]. Then pnx ⩽ x, so (I ⊗ Ψ (pnx)) ⩽ (I ⊗ Ψ )(x), since I ⊗ Ψ is the pre-adjoint of a positive map. In particular, I ⊗ Ψ (pnx) is bounded. By Lemma 2.3 applied to pnx, (I ⊗ Ψ )(x) ⩾ (I ⊗ Ψ )(pnx) ⩾ pnx. (2.11) Since n ⩾ 1 was arbitrary, we conclude from (2.11) that x is bounded, a contradiction which completes the proof. 2 We note for future reference that these results are equally valid for pull down maps of the form Ψ ⊗ I, due to symmetry. These lemmas will be used in Section 4 to derive an important inequality. The next lemma formulates exactly what will be needed.
- 31. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 27 Lemma 2.5. Let Bi ⊆ Mi, i = 1,2, be inclusions of finite von Neumann algebras with pull down maps Ψi. Let B ⊆ M be the inclusion B1 ⊗ B2 ⊆ M1 ⊗ M2. If x ∈ L1( M,eB )+ ∩ M,eB is such that (Ψ1 ⊗ Ψ2)(x) ∈ L1(M) ∩ M and the inequality (Ψ1 ⊗ Ψ2)(x) ⩽ 1 (2.12) is satisfied, then (I ⊗Ψ2)(x) ∈ L1( M1,eB1 ⊗M2)+ ∩( M1,eB1 ⊗M2) and (I ⊗Ψ2)(x) ⩽ 1. Proof. Using the isomorphism of Proposition 2.1(xiii), Ψ1 ⊗ Ψ2 is the pull down map for M,eB . Since (Ψ1 ⊗ I)((I ⊗ Ψ2)(x)) = (Ψ1 ⊗ Ψ2)(x) is a bounded operator by hypothesis, it follows from Lemma 2.4 that (I ⊗ Ψ2)(x) is also bounded in M1,eB1 ⊗ M2. Thus the three operators x, (I ⊗ Ψ2)(x) and (Ψ1 ⊗ Ψ2)(x) are all bounded, and so we may apply Lemma 2.3 twice to the pull down maps I ⊗ Ψ2 and Ψ1 ⊗ I to obtain (Ψ1 ⊗ Ψ2)(x) ⩾ (I ⊗ Ψ2)(x) ⩾ x. (2.13) The result then follows from (2.13) and the hypothesis (2.12). 2 In the proof of Lemma 4.6, we will need the following fact regarding inclusions of finite von Neumann algebras B ⊂ M with B ∩ M ⊆ B. Here, and elsewhere in the paper, we consider inclusions induced by cut-downs. Recall that if Q ⊆ N is an inclusion of von Neumann algebras and q is a projection in Q, then Q ∩ N q = (qQq) ∩ (qNq), Z Q ∩ N q = Z (qQq) ∩ (qNq) , (2.14) see, for example, [18, Section 5.4]. Lemma 2.6. Let B ⊆ M be a containment of finite von Neumann algebras such that B ∩M ⊆ B. If p ∈ M is a nonzero projection, then there exists a nonzero projection q ∈ B which is equivalent to a subprojection of p. Observe that if M is a finite factor, then Lemma 2.6 is immediate. Our proof of Lemma 2.6 is classical, proceeding by analysing the center-valued trace on P . Alternatively one can establish the lemma by taking a direct integral over the center. Since we have been unable to find this fact in the literature we give the details for completeness. Proof of Lemma 2.6. Let denote the center-valued trace on M. We will make use of two properties of from [12, Theorem 8.4.3]. The first is that p1 p2 if and only if (p1) ⩽ (p2), and the second is that p1 ∼ p2 if and only if (p1) = (p2). The hypothesis B ∩ M ⊆ B implies that B ∩ M = Z(B) and, in particular, that Z(M) ⊆ Z(B). For some sufficiently small c 0, the spectral projection z of (p) for the interval [c,1] is nonzero, and (pz) ⩾ cz. Since Bz ⊆ Mz also satisfies the relative commutant hypothesis, it suffices to prove the result under the additional restriction (p) ⩾ c1 for some constant c 0. Let n ⩾ c−1 be any integer. Suppose that it is possible to find a nonzero projection q ∈ B and an orthogonal set {q,p2,...,pn} of equivalent projections in M. The sum of these projections has central trace equal to n (q) and is also bounded by 1, so that (q) ⩽ n−11 ⩽ c1. But then q p and we are done. Thus we may assume that there is an absolute bound on the length of
- 32. 28 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 any such set, and we may then choose one, {q1,p2,...,pn}, of maximal length. By cutting by the central support of q1, we may assume that this central support is 1. Now consider the inclusion q1Bq1 ⊆ q1Mq1, and note that (q1Bq1) ∩ q1Mq1 = q1 B ∩ M = q1Z(B) = Z(q1Bq1) (2.15) from (2.14). Let f1 and f2 be nonzero orthogonal projections in q1Bq1 and q1Mq1 respectively. By the comparison theory of projections, there exists a projection z ∈ Z(q1Mq1) ⊆ Z(q1Bq1) so that zf1 zf2, (1 − z)f2 (1 − z)f1, (2.16) the equivalence being taken in q1Mq1. Now zf1 ∈ q1Bq1 and is equivalent to a subprojec- tion p0 of zf2 ⩽ q1. Then the pair zf1,p0 is equivalent to orthogonal pairs below each pi, 2 ⩽ i ⩽ n, which will contradict the maximal length of {q1,p2,...,pn} unless zf1 = 0. Sim- ilarly (1 − z)f2 = 0. Thus f1 and f2 have orthogonal central supports in q1Bq1 and so [18, Lemma 5.5.3] shows that q1Bq1 is abelian. Eq. (2.15) then shows that q1Bq1 is a masa in q1Mq1, and so another application of [18, Lemma 5.5.3] shows that q1Mq1 is also abelian. Thus q1Bq1 = q1Mq1. Now the projection 1 − q1 − p2 − ··· − pn must be 0, otherwise it would have a nonzero subprojection equivalent to a nonzero projection q̃1 ∈ q1Mq1 = q1Bq1, since q1 has central sup- port 1, and q̃1 would lie in a set of n + 1 equivalent orthogonal projections. Thus q1,p2,...,pn are abelian projections in M with sum 1, so M is isomorphic to L∞(Ω) ⊗ Mn for some measure space Ω. Identify p and q1 with measurable Mn-valued functions. Since q1 is abelian, the rank of q1(ω) is 1 almost everywhere, and the rank of p(ω) is at least 1 almost everywhere since (p) ⩾ c1. Then q1 is equivalent to a subprojection of p since (q1) ⩽ (p). This completes the proof. 2 We conclude this section with a brief explanation of an averaging technique in finite von Neumann algebras which we will use subsequently. It has its origins in [2], but is also used extensively in [14]. If η ∈ L2(M) and U is a group of unitaries in M then the vector can be averaged over U. This is normally associated with amenable groups, but can be made to work in this setting without this assumption. Form the · 2-norm closure K of K = conv uηu∗ : u ∈ U . There is a unique vector η̃ ∈ K of minimal norm, and uniqueness of η̃ implies that uη̃u∗ = η̃ for all u ∈ U. We refer to η̃ as the result of averaging η over U, and many variations of this are possible. We give an example of this technique by establishing a technical result which will be needed in the proof of Theorem 4.7. Recall (Proposition 2.1(xii)) that MeBM is · 2,Tr-dense in L2( M,eB ,Tr). Consider a ∗- subalgebra A which is strongly dense in M. If x,y ∈ M, then fix sequences {xn}∞ n=1, {yn}∞ n=1 from A converging strongly to x and y, respectively. Then (x − xn)eB 2 2,Tr = Tr eB(x − xn)∗ (x − xn)eB = τ (x − xn)∗ (x − xn) = (x − xn)ξ,(x − xn)ξ , (2.17)
- 33. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 29 so xneB → xeB in · 2,Tr-norm. Thus xneBy → xeBy so, given ε 0, we may choose n0 so large that xn0 eBy − xeBy 2,Tr ε/2. The same argument on the right allows us to choose n1 so large that xn0 eBy − xn0 eByn1 2,Tr ε/2, whereupon xeBy − xn0 eByn1 2,Tr ε. The conclusion reached is that the algebra AeBA = { n i=1 xieByi: xi,yi ∈ A} is · 2,Tr-norm dense in L2( M,eB ,Tr). In the next lemma, we will use this when M is a tensor product M1 ⊗ M2 where we take A to be the algebraic tensor product M1 ⊗ M2. Lemma 2.7. Let B1,B2 be von Neumann subalgebras of finite von Neumann algebras M1,M2 and let B = B1 ⊗ B2, M = M1 ⊗ M2. Then L2 Z B ∩ M,eB ,Tr = L2 Z B 1 ∩ M1,eB1 ,Tr1 ⊗2 L2 Z B 2 ∩ M2,eB2 ,Tr2 . Note that, although Tri is a semifinite trace on Mi,eBi , it need not be semifinite on Z(B i ∩ Mi,eBi ). This is why the lemma cannot be obtained immediately from the uniqueness of product measures on σ-finite measure spaces. Proof of Lemma 2.7. If zi ∈ Z(B i ∩ Mi,eBi ), i = 1,2, then z1 ⊗ z2 ∈ Z(B ∩ M,eB ) and z1 ⊗ z2 2,Tr = z1 2,Tr1 z2 2,Tr2 . This shows the containment from right to left. Suppose that z ∈ Z(B ∩ M,eB ) with Tr(z∗z) ∞. Then z lies in L2( M,eB ,Tr) so can be approximated in · 2,Tr-norm by sums of the form k i=1 xieByi with xi,yi ∈ M1 ⊗ M2. The preceding remarks then allow us to assume that xi and yi lie in the algebraic tensor product M1 ⊗ M2. Thus, given ε 0, we may find elements ai,ci ∈ M1, bi,di ∈ M2 so that z − n i=1 (ai ⊗ bi)(eB1 ⊗ eB2 )(ci ⊗ di) 2,Tr ⩽ ε. (2.18) This may be rewritten as z − n i=1 (aieB1 ci) ⊗ (bieB2 di) 2,Tr ⩽ ε, (2.19) and then as z − n i=1 fi ⊗ gi 2,Tr ⩽ ε, (2.20) where fi ∈ M1eB1 M1 and gi ∈ M2eB2 M2. We may further suppose that the set {g1,...,gn} is linearly independent. For j = 1,2, let Nj be the von Neumann algebra generated by Bj and B j ∩ Mj ,eBj , and note that z commutes with N1 ⊗ N2. Let K = convw n i=1 ufiu∗ ⊗ gi: u ∈ U(N1) , Ki = convw ufiu∗ : u ∈ U(N1)
- 34. 30 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 for 1 ⩽ i ⩽ n. Then K ⊆ n i=1 Ki ⊗ gi. By [18, Lemma 9.2.1] K and each Ki are closed in their respective · 2-norms. If k ∈ K is the element of minimal · 2-norm in K then it may be written as k = n i=1 ki ⊗ gi with ki ∈ Ki. Since k is invariant for the action of U(N1 ⊗ 1), we see that n i=1 ukiu∗ − ki ⊗ gi = 0, u ∈ U(N1). (2.21) The linear independence of the gi’s allows us to conclude that ukiu∗ = ki for 1 ⩽ i ⩽ n and u ∈ U(N1). Thus ki ∈ N 1 ∩ M1,eB1 = Z(B 1 ∩ M1,eB1 ). The inequality (2.20) is preserved by averaging in this manner over U(N1 ⊗1) so, replacing each fi by ki if necessary, we may assume that fi ∈ Z(B 1 ∩ M1,eB1 ) for 1 ⩽ i ⩽ n. Now repeat this argument on the right, averaging over U(1⊗N2), to replace the gi’s by elements of Z(B 2 ∩ M2,eB2 ). With these changes, (2.20) now approximates z by a sum from L2 Z B 1 ∩ M1,eB1 ,Tr1 ⊗2 L2 Z B 2 ∩ M2,eB2 ,Tr2 which proves the containment from left to right and establishes equality. 2 3. Projections in the basic construction In this section, we relate intertwiners of a subalgebra to certain projections in the basic con- struction. We consider a finite von Neumann algebra M and a von Neumann subalgebra B whose unit will always coincide with that of M. For the most part, we will be interested in the condition B ∩ M ⊆ B (equivalent to B ∩ M = Z(B)), but we will make this requirement explicit when it is needed. Lemma 3.1. Let B be a von Neumann subalgebra of a finite von Neumann algebra M and let v ∈ GN(1) (B). 1. Then v∗eBv is a projection in (B ∩ M,eB )v∗v. 2. Suppose q is a projection in B. Then v∗eBv lies in (B ∩ M,eB )q if, and only if, v∗v ∈ Z(B)q = Z(qBq). Proof. 1. The element v∗eBv is positive in M,eB . Since vv∗ ∈ B and so commutes with eB, the following calculation establishes that v∗eBv is a projection: v∗ eBv 2 = v∗ eBvv∗ eBv = v∗ vv∗ eBv = v∗ eBv. (3.1) For an arbitrary b ∈ v∗vBv∗v, v∗ eBv b = v∗ eBvbv∗ v = v∗ vbv∗ eBv = v∗ vbv∗ vv∗ eBv = b v∗ eBv , (3.2) where the second equality uses vbv∗ ∈ B to commute this element with eB. Thus (3.2) establishes that v∗eBv ∈ (v∗vBv∗v) ∩ v∗v M,eB v∗v which is (B ∩ M,eB )v∗v (see (2.14)).
- 35. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 31 2. Suppose now that v∗v ∈ Z(B)q = Z(qBq). It is immediate that Z(qBq) ⊆ Z((B ∩ M,eB )q), so we have a decomposition B ∩ M,eB q = B ∩ M,eB v∗ v ⊕ B ∩ M,eB q − v∗ v . (3.3) We have already shown that v∗eBv is in the first summand of (3.3) so must lie in (B ∩ M,eB )q. Conversely, the hypothesis on v∗eBv implies that v∗eBv = v∗eBvq = qv∗eBv, so the pull down map gives v∗v = v∗vq = qv∗v, showing that v∗v ∈ qBq. For each b ∈ B, v∗eBvqbq = qbqv∗eBv. Applying the pull down map gives v∗vqbq = qbqv∗v and hence v∗v ∈ Z(qBq). 2 We now strengthen this lemma under the additional hypothesis that B ∩ M ⊆ B. Recall from part (iv) of Proposition 2.1 that JZ(B)J = Z( M,eB ). When B ⊂ M is a finite index inclusion of irreducible subfactors, Lemma 3.2 is contained in [13, Propositions 1.7(2) and 1.9]. The proof follows the extension to infinite index inclusions of irreducible subfactors in [20, Lemma 3.3]. Lemma 3.2. Let B be a von Neumann subalgebra of a finite von Neumann algebra M and suppose that B ∩ M ⊆ B. Let v ∈ GN(1) (B). Then the projection v∗eBv is central in (B ∩ M,eB )v∗v. Proof. Define two projections p,q ∈ B by p = v∗v and q = vv∗. Now consider an arbitrary x ∈ (B ∩ M,eB )p = (pBp) ∩ p M,eB p. Then, for each b ∈ B, vxv∗ vbv∗ = vxpbpv∗ = vpbpxv∗ = vbv∗ vxv∗ , (3.4) showing that vxv∗ ∈ (vBv∗) ∩ (q M,eB q). We next prove that qeB is central in (vBv∗) ∩ (q M,eB q). It lies in this algebra by the previous calculation and Lemma 3.1, as qeB = v(v∗eBv)v∗. Take t ∈ (vBv∗) ∩ (q M,eB q) to be self-adjoint and let η be tξ ∈ L2(M). Now take a sequence {xn}∞ n=1 from M converging in · 2-norm to tξ. Since t = qt = tq, we may assume that the sequence {xn}∞ n=1 lies in qMq, otherwise replace it by {qxnq}∞ n=1. For each u in the unitary group U(pBp),t commutes with vuv∗ and so Jvuv∗ Jvuv∗ η = Jvuv∗ Jvuv∗ tξ = Jvuv∗ Jtvuv∗ ξ = tJvuv∗ Jvuv∗ ξ = tvuv∗ vu∗ v∗ ξ = tqξ = tξ = η, (3.5) where the third equality holds because vuv∗ ∈ B so that Jvuv∗J ∈ ( M,eB ). For each n ⩾ 1 and each u ∈ U(pBp), Jvuv∗ Jvuv∗ xnξ − η 2 = Jvuv∗ Jvuv∗ (xnξ − η) 2 ⩽ xnξ − η 2, (3.6) from (3.5). If we let yn be the element of qMq obtained by averaging xn over the unitary group vU(pBp)v∗ ⊆ qBq, then (3.6) gives ynξ − η 2 ⩽ xnξ − η 2, n ⩾ 1, (3.7)
- 36. 32 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 while yn ∈ (vBv∗) ∩ qMq. Since vBv∗ ∩ qMq = vBv∗ ∩ vMv∗ = v B ∩ M v∗ = vZ(B)v∗ ⊆ qBq, (3.8) we see that yn ∈ qBq for n ⩾ 1. From (3.7) it follows that η ∈ L2(qBq). For b ∈ B, tqbξ = tJb∗ qJξ = Jb∗ qJtξ = lim n→∞ Jb∗ qJynξ = lim n→∞ ynqbξ, (3.9) showing that tqbξ ∈ L2(qB). Thus L2(qB) is an invariant subspace for t. The projection onto it is qeB, so tqeB = qeBtqeB. Since t is self-adjoint, we obtain that qeB commutes with (vBv∗) ∩ (q M,eB q), establishing centrality. It was established in Eq. (3.4) that vxv∗ ∈ (vBv∗) ∩ (q M,eB q) whenever x ∈ (B ∩ M,eB )p, and so each such vxv∗ commutes with qeB. Thus v∗ eBvx = v∗ qeBvxv∗ v = v∗ vxv∗ qeBv = xv∗ eBv (3.10) for x ∈ (B ∩ M,eB )p, showing that v∗eBv is central in (B ∩ M,eB )p. 2 Since the centrality of the projections v∗eBv in (B ∩ M,eB )v∗v is crucial to our subsequent arguments, the following corollary highlights why the hypothesis B ∩ M ⊆ B is essential. Corollary 3.3. Let B be a von Neumann subalgebra of a finite von Neumann algebra M. Then eB is central in B ∩ M,eB if and only if B ∩ M ⊆ B. Proof. If B ∩ M ⊆ B, then centrality of eB is a special case of Lemma 3.2 with v = 1. Con- versely, suppose that eB is central in B ∩ M,eB and consider x ∈ B ∩ M. Then x commutes with eB so x ∈ B by Proposition 2.1(iii). Thus B ∩ M ⊆ B. 2 Lemma 3.4. Let B be a von Neumann subalgebra of a finite von Neumann algebra M, suppose that B ∩ M ⊆ B, and let v ∈ GN(1) (B). Then any subprojection of v∗eBv in (B ∩ M,eB )v∗v has the form pv∗eBv where p is a central projection in B. Proof. By Lemma 3.1(i), v∗eBv ∈ (B ∩ M,eB )v∗v. Suppose that a projection q ∈ (B ∩ M,eB )v∗v lies below v∗eBv. Then (vqv∗)2 = vqv∗vqv∗ = vqv∗, so vqv∗ is a projection. The relation vqv∗ vbv∗ = vq v∗ vbv∗ v v∗ = v v∗ vbv∗ v qv∗ = vbv∗ vqv∗ , b ∈ B, (3.11) shows that vqv∗ ∈ (vBv∗) ∩ vv∗ M,eB vv∗. Moreover, vqv∗ ⩽ vv∗eBvv∗ = eBvv∗, so there exists a projection f ∈ vv∗Bvv∗ such that vqv∗ = f eB. For b ∈ B, f vbv∗ eB = f eBvbv∗ = vqv∗ vbv∗ = vbv∗ vqv∗ = vbv∗ f eB, (3.12)
- 37. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 33 and so f vbv∗ = vbv∗f . Thus f ∈ vBv∗ ∩ vv∗ Bvv∗ . (3.13) If b0 ∈ B is such that vv∗b0vv∗ commutes with vBv∗, then vv∗ b0vv∗ vbv∗ = vbv∗ vv∗ b0vv∗ , b ∈ B. (3.14) Multiply on the left by v∗ and on the right by v to obtain v∗ b0vv∗ vbv∗ v = v∗ vbv∗ vv∗ b0v, b ∈ B. (3.15) Thus v∗ b0v ∈ v∗ vBv∗ v ∩ v∗ vMv∗ v = v∗ v B ∩ M v∗ v = v∗ vZ(B)v∗ v. (3.16) Consequently, vv∗b0vv∗ ∈ vZ(B)v∗. It follows that (vBv∗) ∩ vv∗Bvv∗ ⊆ vZ(B)v∗, and so there is a central projection p ∈ Z(B) so that f = vpv∗. We now have q = v∗ f eBv = v∗ vpv∗ eBv = pv∗ eBv, (3.17) as required. 2 In Section 4, we wish to use the projection v∗eBv to investigate an intertwiner v of a ten- sor product B = B1 ⊗ B2 ⊂ M1 ⊗ M2 = M, where each B i ∩ Mi ⊆ Bi. In conjunction with Proposition 2.1(xiii), Tomita’s commutation theorem gives B ∩ M,eB ∼ = B 1 ∩ M1,eB1 ⊗ B 2 ∩ M2,eB2 . (3.18) By Lemma 3.2, such an intertwiner gives rise to a central projection v∗eBv in (B ∩ M,eB )v∗v. Unfortunately, in general the projection v∗v will not factorize as an elementary tensor of projec- tions b1 ⊗ b2, with bi ∈ Bi, and so the algebra (B ∩ M,eB )v∗v will not decompose as a tensor product. This prevents us from applying tensor product techniques to the projection v∗eBv di- rectly. However, standard von Neumann algebra theory (see, for example, [11]) gives a central projection P ∈ B ∩ M,eB such that v∗eBv = Pv∗v. Since we need to ensure that the projec- tion P fully reflects the properties of v, we cannot just appeal to the general theory to obtain P , so we give an explicit construction in Definition 3.5 below. The subsequent lemmas set out the properties of P that we require later. Definition 3.5. Let B ⊂ M be an inclusion of finite von Neumann algebras with B ∩ M ⊆ B and let v ∈ GN (1) M (B). Let z ∈ Z(B) be the central support of v∗v. Define p0 to be v∗v, and let {p0,p1,...} be a family of nonzero pairwise orthogonal projections in B which is max- imal with respect to the requirements that pn ⩽ z and each pn is equivalent in B to a sub- projection in B of p0. Since two projections in a von Neumann algebra with non-orthogonal central supports have equivalent nonzero subprojections, maximality gives n⩾0 pn = z. For n ⩾ 1, choose partial isometries wn ∈ B so that w∗ nwn = qn ⩽ p0 and wnw∗ n = pn. Then define
- 38. 34 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 vn = vw∗ n ∈ GN (1) M (B). Lemma 3.1 shows that v∗ neBvn ∈ (B ∩ M,eB )v∗ nvn and this space is (B ∩ M,eB )pn since v∗ nvn = wnv∗vw∗ n = pn. In particular, {v∗ neBvn}n⩾0 is a set of pairwise orthogonal projections so we may define a projection Pv = n⩾0 v∗ neBvn in M,eB . Lemma 3.6. With the notation of Definition 3.5, the projection Pv is central in B ∩ M,eB and satisfies Pvv∗v = v∗eBv. Proof. This projection is Pv = n⩾0 wnv∗eBvw∗ n. By Lemma 3.2 there exists t ∈ Z(B ∩ M,eB ) so that v∗eBv = tv∗v, and so Pv becomes Pv = n⩾0 wntv∗ vw∗ n = n⩾0 twnv∗ vw∗ n = n⩾0 twnp0w∗ n = tz. (3.19) Thus Pv ∈ B ∩ M,eB and Pvv∗v = tv∗v = v∗eBv since z is the central support of v∗v. Since z ∈ Z(B) ⊂ Z(B ∩ M,eB ), it follows that Pv = tz also lies in Z(B ∩ M,eB ). 2 Remark 3.7. The proof of Lemma 3.6 shows that Pv is the minimal projection in B ∩ M,eB with Pvv∗v = v∗eBv. This gives a canonical description of Pv which is independent of the choices made in Definition 3.5. The explicit formulation of the definition is useful in transferring properties from v∗eBv to Pv. We now identify the subprojections of Pv. This will be accomplished by the next lemma, which considers a wider class of projections needed subsequently. Let {vi}∞ i=1 be a sequence from GN(1) (B) satisfying viv∗ j = 0 for i = j, let p ∈ B be the projection ∞ i=1 v∗ i vi and let P ∈ M,eB be the projection ∞ i=1 v∗ i eBvi. In particular, the projection Pv of Definition 3.5 is of this form. Let N(P) denote the von Neumann algebra N(P) = {x ∈ pMp: xP = Px} ⊆ M. (3.20) Lemma 3.8. Let P = ∞ i=1 v∗ i eBvi be as above, and let p = ∞ i=1 v∗ i vi ∈ B. (i) If x ∈ pMp satisfies xP = 0, then x = 0; (ii) The map x → xP is a ∗-isomorphism of N(P) into M,eB ; (iii) A projection Q ∈ M,eB satisfies Q ⩽ P if and only if there exists a projection f ∈ N(P) such that Q = f P . Moreover, if P and Q lie in B ∩ M,eB , then f ∈ Z(B) and Q has the same form as P . Proof. (i) Suppose that x ∈ pMp and xP = 0. Then x ∞ i=1 v∗ i eBvi = 0. (3.21) Multiply on the right in (3.21) by v∗ k vk to obtain xv∗ k eBvk = 0 for k ⩾ 1. The pull down map gives xv∗ k vk = 0. Summing over k shows that xp = 0 and the result follows since x = xp. (ii) Since P ∈ N(P), the map x → xP is a ∗-homomorphism on N(P). It has trivial kernel, by (i), so is a ∗-isomorphism.
- 39. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 35 (iii) If f ∈ N(P), then it is clear that f P is a projection below P , since f commutes with P . Conversely, consider a projection Q ⩽ P with Q ∈ M,eB . The introduction of partial sums below is to circumvent some questions of convergence. Define Pk = k i=1 v∗ i eBvi. Then limk→∞ Pk = P strongly, so PkQPk converges strongly to PQP = Q. For m,n ⩾ 1, let bm,n ∈ B be the element such that bm,neB = eBvmQv∗ neB. Then PkQPk = k m,n=1 v∗ meBvmQv∗ neBvn = k m,n=1 v∗ mbm,neBvn = k m,n=1 v∗ mbm,neBvnv∗ nvn = k m,n=1 v∗ mbm,nvnv∗ neBvn. (3.22) Now define xk ∈ pW∗(GN(1) (B))p by xk = k m,n=1 v∗ mbm,nvn. The relations viv∗ j = 0 for i = j allow us to verify that xkP = PkQPk, and consequently xkP = Px∗ k , k ⩾ 1, (3.23) since PkQPk is self-adjoint. Thus ∞ i=1 xkv∗ i eBvi = ∞ i=1 v∗ i eBvix∗ k . (3.24) The sums in (3.24) converge in · 1-norm, so we may apply the pull down map to obtain xkp = px∗ k . Since xk = xkp, we conclude that xk is self-adjoint. Thus, from (3.23), xk commutes with P , and so lies in N(P). From above, xkP ⩾ 0 and xkP ⩽ 1, so xk ⩾ 0 and xk ⩽ 1 by (ii). Let f be a σ-weak accumulation point of the sequence {xk}∞ k=1. Then f ⩾ 0, f ⩽ 1, and f ∈ N(P). Since xkP = PkQPk, we conclude that f P = Q. It now follows from (ii) that f is a projection in N(P). If P ∈ B ∩ M,eB , then the pull down map gives bp = pb for b ∈ B, so p ∈ Z(B). If also Q ∈ B ∩ M,eB , then Q = f P and f ⩽ p. If b ∈ Bp then commutation with Q gives (bf − f b)P = 0, so f ∈ B ∩ M = Z(B), by (i). Finally Q = f P = i(vif )∗eB(vif ), so is of the same form as P . 2 4. Intertwiners of tensor products In this section we will prove one of our main results, the equality of W∗(GN(1) (B1)) ⊗ W∗(GN(1) (B2)) and W∗(GN(1) (B1 ⊗B2)), where Bi ⊆ Mi, i = 1,2, are inclusions of finite von Neumann algebras satisfying B i ∩ Mi ⊆ Bi. The key theorem for achieving this is the following one, which enables us to detect those central projections in corners of the relative commutant of the basic construction which arise from intertwiners. It is inspired by [1, Proposition 2.7], although is not a direct generalization of that result. For comparison, [1, Proposition 2.7] shows that, in the case of a masa A, a projection P ∈ A ∩ M,eA which is subequivalent to eA dom- inates an operator v∗eBv for some v ∈ GN(A). Example 4.3 below will show that such a result will not hold in general without additional hypotheses.
- 40. 36 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 Theorem 4.1. Let A be an abelian von Neumann algebra with a fixed faithful normal semifinite weight Φ. Let B be a von Neumann subalgebra of a finite von Neumann algebra M with a faithful normal trace τ satisfying B ∩ M ⊆ B. Fix a projection q ∈ A ⊗ B and suppose that P ∈ (A ⊗ (B ∩ M,eB ))q is a nonzero projection such that P (1 ⊗ eB) in A ⊗ M,eB , and satisfies (Φ ⊗ Tr)(Pr) ⩽ (Φ ⊗ τ)(qr) (4.1) for all projections r ∈ Z(q(A ⊗ B)q). Then there exists an element v ∈ GN (1) A⊗M (A ⊗ B) such that P = v∗(1 ⊗ eB)v. Before embarking on the proof, let us recall that, for a finite von Neumann algebra M with a faithful normal trace τ, we regard the Hilbert space L2(M) as the completion of M in the norm x 2,τ = τ(x∗x)1/2 and L1(M) as the completion of M in the norm x 1,τ = τ(|x|). The Cauchy–Schwarz inequality gives xy∗ 1,τ ⩽ x 2,τ y 2,τ for x,y ∈ M and so this inequality allows us to define ζη∗ ∈ L1(M) for ζ,η ∈ L2(M). In particular, if (yn) is a sequence in M converging to η ∈ L2(M), then y∗ nyn → η∗η in L1(M). Recall too that we can regard elements of L2(M) as unbounded operators on L2(M) affiliated to M. The only fact we need about these unbounded operators is that if η ∈ L2(M) satisfies η∗η ∈ M (regarded as a subset of L1(M)), then in fact η ∈ M. This follows as η has a polar decomposition v(η∗η)1/2, where v is a partial isometry in M and (η∗η)1/2 is an element of L2(M), which lies in M if η∗η does. Proof of Theorem 4.1. The first case that we will consider is where A = C and Φ is the identity map. Then the hypothesis becomes Tr(Pr) ⩽ τ(qr) (4.2) for all projections r ∈ Z(B)q. Since P eB, there exists a partial isometry V ∈ M,eB such that P = V ∗V and V V ∗ ⩽ eB. Define the map θ: qBq → BeB by θ(qbq) = V qbqV ∗ = eBV qbqV ∗ eB, b ∈ B. (4.3) Then θ is a ∗-homomorphism since V ∗V commutes with qBq, and so there is a ∗-homomor- phism φ : qBq → B so that θ(qBq) = φ(qbq)eB for qbq ∈ qBq. Thus qbqV ∗ = qbqV ∗ V V ∗ = V ∗ V qbqV ∗ = V ∗ eBφ(qbq) = V ∗ φ(qbq) (4.4) for qbq ∈ qBq. Now define η ∈ L2(M) by η = JV ∗ξ, and observe that η = 0 since V Jη = V V ∗ξ = b0ξ, where V V ∗ = b0eB for some b0 ∈ B. If we apply (4.4) to ξ, then the result is qbqJη = V ∗ φ(qbq)ξ = V ∗ Jφ qb∗ q Jξ = Jφ qb∗ q JV ∗ ξ = Jφ qb∗ q η, qbq ∈ qBq, (4.5)
- 41. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 37 where we have used M,eB = (JBJ) to commute Jφ(qb∗q)J with V ∗. Multiply (4.10) on the left by J and replace b by b∗ to obtain ηqbq = φ(qbq)η, qbq ∈ qBq. (4.6) Taking b = 1, (4.5) becomes JqJη = φ(q)η, (4.7) so JqV ∗ ξ = φ(q)η. (4.8) Multiply on the left by V J to obtain φ(q)ξ = V Jφ(q)η, (4.9) showing that φ(q)η = 0. From (4.7), φ(q)ηq = φ(q)η = 0, and this allows us to assume in (4.6) that the vector η is nonzero and satisfies ηq = η, φ(q)η = η by replacing η with φ(q)ηq if necessary. For unitaries u ∈ qBq, (4.6) becomes φ u∗ ηu = φ u∗ φ(u)η = φ(q)η = η. (4.10) Choose a sequence {xn}∞ n=1 from M such that xnξ → η in · 2,τ -norm. Since φ(q)ηq = η, we may assume that φ(q)xnq = xn for n ⩾ 1. Let yn be the element of minimal · 2,τ -norm in convw {φ(u∗)xnu: u ∈ U(qbq)}. Since φ u∗ xnuξ − η 2,τ = φ u∗ (xnξ − η)u 2,τ ⩽ xnξ − η 2,τ (4.11) for all u ∈ U(qBq), we see that ynξ − η 2,τ ⩽ xnξ − η 2,τ , so ynξ → η in · 2,τ -norm and φ(u∗)ynu = yn for n ⩾ 1 by the choice of yn. Then ynu = φ(u)yn for u ∈ U(qBq), so ynqbq = φ(qbq)yn, n ⩾ 1, qbq ∈ qBq. (4.12) Thus y∗ nynqbq = y∗ nφ(qbq)yn, n ⩾ 1, qbq ∈ qBq, (4.13) and this implies that y∗ nyn ∈ (qBq) ∩ qMq = (B ∩ M)q = Z(B)q for each n ⩾ 1. The discus- sion preceding the proof ensures that y∗ nyn → η∗η in L1(M), so we see that η∗η ∈ L1(Z(B)q). For each z ∈ Z(B)q, τ η∗ ηzq = lim n→∞ τ y∗ nynqz = lim n→∞ ynqzξ,ynξ = Jz∗ qJη,η = Jz∗ qJJV ∗ ξ,JV ∗ ξ = z∗ qV ∗ ξ,V ∗ ξ = V z∗ qV ∗ ξ,ξ = φ z∗ q ξ,ξ = τ φ(zq) . (4.14)
- 42. 38 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 Now Tr(Pzq) = Tr V ∗ V zq = Tr V zqV ∗ = Tr φ(zq)eB = τ φ(zq) . (4.15) Thus, from (4.14), (4.15) and the hypothesis (4.2), τ η∗ ηr = Tr(Pr) ⩽ τ(r) (4.16) for all projections r ∈ Z(B)q. Since Z(B)q is abelian, simple measure theory allows us to conclude from (4.16) that η∗η ∈ Z(B)q (rather than just L1(Z(B)q)) and so η ∈ M, by the discussion prior to the start of the proof. Moreover, (4.16) also gives η ⩽ 1, by taking r to be the spectral projection of η∗η for the interval (c,∞) where c 1 is arbitrary. Since η ∈ L2(M) has been proved to lie in M, we rename this nonzero operator as x ∈ M. From above, x ⩽ 1 and x∗x ∈ Z(B)q. Since JV ∗ξ = η = xξ = xJξ, for y ∈ M and b ∈ B, (V − x)yξ,bξ = yξ,V ∗ Jb∗ Jξ − xyξ,bξ = yξ,Jb∗ JV ∗ ξ − xyξ,bξ = yξ,Jb∗ xJξ − xyξ,bξ = yξ,x∗ bξ − xyξ,bξ = 0, (4.17) and so eBV = eBx, implying that V = eBx. Thus P = V ∗V = x∗eBx. Since P2 = P , x∗ eBx = x∗ eBxx∗ eBx = x∗ EB xx∗ eBx, (4.18) so the pull down map gives x∗ x = x∗ EB xx∗ x. (4.19) This equation is x∗(1 − EB(xx∗))x = 0, so (1 − EB(xx∗))1/2x = 0. Thus EB(xx∗)x = x. If we multiply on the right by x∗ and apply EB, then we conclude that EB(xx∗) is a projection. Moreover, EB xx∗ xx∗ = xx∗ , (4.20) so EB(xx∗) ⩾ xx∗ since x ⩽ 1. The trace then gives equality, and so x is a partial isometry with x∗x, xx∗ ∈ B. Since x = xq and x∗eBx = P , which commutes with qBq, we obtain xbx∗ eB = xqbqx∗ xx∗ eB = xqbqx∗ eBxx∗ = xx∗ eBxqbqx∗ = eBxbx∗ , b ∈ B, (4.21) showing that xBx∗ ⊆ B. Thus x ∈ GN(1) (B) and P = x∗eBx. This completes the proof when A = C and Φ is the identity map.
- 43. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 39 The second case is when A is an arbitrary abelian von Neumann algebra, and Φ is bounded, so we may assume that Φ is a state on A since (4.1) is unaffected by scaling. The result now follows from the first case by replacing the inclusion B ⊆ M by A ⊗ B ⊆ A ⊗ M. The trace on A ⊗ M is Φ ⊗ τ and eA⊗B is 1 ⊗ eB, so the canonical trace on A ⊗ M,eA⊗B is Φ ⊗ Tr. The last case is where Φ is a faithful normal semifinite weight. Then there is a family {fλ}λ∈Λ of orthogonal projections in A with sum 1 such that Φ(fλ) ∞ for each λ ∈ Λ. If Φλ denotes the restriction of Φ to Afλ, then Φλ is bounded. If we replace P , A, Φ and q by respectively P(fλ ⊗ 1), Afλ, Φλ and q(fλ ⊗ 1), then we are in the second case. Thus there exists, for each λ ∈ Λ, a partial isometry vλ ∈ GN (1) Afλ⊗M (Afλ ⊗ B) so that P(fλ ⊗ 1) = v∗ λ(eAfλ⊗B)vλ. The central support of vλ lies below fλ ⊗ 1 so we may define v ∈ GN (1) A⊗M (A ⊗ B) by v = λ∈Λ vλ, and it is routine to check that P = v∗eBv. 2 Theorem 4.1 characterizes those projections in the basic construction which arise from inter- twiners. Corollary 4.2. Given an inclusion B ⊆ M of finite von Neumann algebras with B ∩ M ⊆ B and a projection q ∈ B, a projection P ∈ (B ∩ M,eB )q is of the form v∗eBv for some intertwiner v ∈ GN(1) (B) if and only if P eB in M,eB and Tr(Pr) ⩽ τ(qr) for all projections r ∈ Z(B)q. Furthermore in this case the domain projection v∗v must lie in Z(B)q. Proof. Taking A = C and Φ to be the identity in Theorem 4.1 shows that any projection P sat- isfying the conditions of the corollary is of the form v∗eBv for some intertwiner v ∈ GN(1) (B). Lemma 3.1(ii) then shows that v∗v ∈ Z(B)q. Conversely, given an intertwiner v ∈ GN(1) (B), Lemma 3.1(ii) shows that v∗eBv ∈ (B ∩ M,eB )q precisely when v∗v ∈ Z(B)q. The other two conditions of the corollary follow as v∗eBv ∼ vv∗eB ⩽ eB in M,eB , and for a projection r ∈ Z(B)q, Tr(v∗eBvr) = τ(v∗vr) ⩽ τ(qr). 2 The tracial hypothesis (4.1) of Theorem 4.1 is an extra ingredient in this theorem as compared with [1, Proposition 2.7]. The following example shows that Theorem 4.1 can fail without this hypothesis. Example 4.3. Let R be the hyperfinite II1 factor and fix an outer automorphism θ of period two. Let M = M3 ⊗ R and let B = a 0 0 0 b c 0 θ(c) θ(b) : a,b,c ∈ R ⊆ M. Note that B ∼ = R ⊕ (R θ Z2). It is straightforward to verify that B ∩ M = Z(B) = Ce11 ⊕ C(e22 + e33) where {ei,j }3 i,j=1 are the matrix units. Let P ∈ M,eB be ( √ 2e12)eB( √ 2e21). Since EB(e22) = (e22 + e33)/2, P is a projection, and it is routine to verify that P commutes with B. If there is a nonzero intertwiner v ∈ M such that v∗eBv ⩽ P , then v = ve11. Direct cal- culation shows that v would then have the form we11 for some partial isometry w ∈ R, so v∗eBv would be qe11eB for some projection q ∈ R. However, this nonzero projection is orthogonal to P and so cannot lie under it. Thus the conclusion of Theorem 4.1 fails in this case. Note that Tr(Pe11) = τ(2e11) = 2/3, while τ(e11) = 1/3, so the tracial hypothesis of Theorem 4.1 is not satisfied.
- 44. 40 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 It is worth noting that B ∩ M,eB can be explicitly calculated in this case. This algebra is abelian and five-dimensional with minimal projections e11eB, (1 − e11)eB, (1 − e11)(1 − eB), ( √ 2e12)eB( √ 2e21), and e21eBe12 + e31eBe13. The corresponding B-bimodules in L2(M) are generated respectively by the vectors e11, e22 + e33, e22 − e33, e12, and e21. For the remainder of the section we fix inclusions Bi ⊆ Mi of finite von Neumann algebras satisfying B i ∩Mi ⊆ Bi for i = 1,2, and we denote the inclusion B1 ⊗B2 ⊆ M1 ⊗M2 by B ⊆ M. For i = 1,2, let Si = sup Pj ∈ Z B i ∩ Mi,eBi : Tri(Pj ) ∞ . Note that Si acts as the identity on L2(Z(B i ∩ Mi,eBi ),Tri). Given v ∈ GN (1) M (B), the pro- jection Pv of Definition 3.5 satisfies Tr(Pv) ⩽ 1, and Pv ∈ Z(B ∩ M,eB ) by Lemma 3.6. It follows from Lemma 2.7 that Pv ⩽ S1 ⊗ S2. Although Tri restricted to Z(B i ∩ Mi,eBi ) might not be semifinite, it does have this property on the abelian von Neumann algebra Ai = Z(B i ∩ Mi,eBi )Si by the choice of Si. Moreover, each Pv is an element of A1 ⊗ M2,eB2 . We need two further projections which we define below. Definition 4.4. For i = 1,2, let Pi denote the collection of projections R ∈ Z(B i ∩ Mi,eBi ) which are expressible as R = n⩾1 v∗ neBi vn, vn ∈ GN(1) (Bi) where {v∗ nvn}n⩾1 is an orthogonal set of projections in Bi. Such a projection satisfies Tri(R) ⩽ 1. Let Qi be the supremum of the projections in Pi, so Qi ⩽ Si. Our next objective is to show that each projection Pv arising from an intertwiner lies below Q1 ⊗ Q2. For the next two lemmas, let v ∈ GN (1) M (B) be fixed. We continue to employ the notation Pv for the projection n v∗ neBvn ∈ Z(B ∩ M,eB ) which satisfies Pvv∗v = v∗eBv. Let A1 be the abelian von Neumann algebra Z(B 1 ∩ M1,eB1 )S1 on which Φ, the restriction of Tr1, is semifinite. Lemma 4.5. If r is a projection in A1 ⊗ Z(B2), then (Φ ⊗ Tr2)(Pvr) ⩽ (Φ ⊗ τ2)(r). (4.22) Proof. There is a measure space (Ω,Σ,μ) so that A1 corresponds to L∞(Ω) while Φ is given by integration with respect to the σ-finite measure μ. Then Pv is viewed as a projection-valued function Pv(ω), with the same representation r(ω) for r. For i = 1,2, let Ψi be the pull down map for Mi,eBi . Then (Ψ1 ⊗ Ψ2)(Pv) = n v∗ nvn which is a projection, so has norm 1. By Lemma 2.5, (I ⊗Ψ2)(Pv) ⩽ 1, and so this element of A1 ⊗M2 can be represented as a function f (ω) with f (ω) ⩽ 1 almost everywhere. It follows that
- 45. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 41 (Φ ⊗ τ2)(r) = Ω τ2 r(ω) dμ(ω) ⩾ Ω τ2 r(ω)f (ω) dμ(ω) = (Φ ⊗ τ2) (I ⊗ Ψ2)(Pv)r = (Φ ⊗ τ2) (I ⊗ Ψ2)(Pvr) = (Φ ⊗ Tr2)(Pvr), (4.23) where the penultimate equality is valid because r ∈ A1 ⊗ Z(B2) and the M2-bimodular property of Ψ2 applies. 2 Lemma 4.6. For v ∈ GN(1) (B), the associated projection Pv ∈ Z(B ∩ M,eB ) satisfies Pv ⩽ Q1 ⊗ Q2. Proof. The remarks preceding Definition 4.4 show that Pv ⩽ S1 ⊗S2. We will prove the stronger inequality Pv ⩽ S1 ⊗ Q2, which is sufficient to establish the result since we will also have Pv ⩽ Q1 ⊗ S2 by a symmetric argument. As before, let A1 denote Z(B 1 ∩ M1,eB1 )S1 and let Φ be the restriction of Tr1 to A1. Then, as noted earlier, A1 is an abelian von Neumann algebra and Φ is a faithful normal semifinite weight on A1. Let {qn}∞ n=1 be a maximal family of nonzero orthogonal projections in Z(B1)⊗B2 so that Pvqn = w∗ n(1 ⊗ eB2 )wn for partial isometries wn ∈ A1 ⊗ M2 which are intertwiners of A1 ⊗ B2. Let q = ∞ n=1 qn, defining q to be 0 if no such projections exist. We will first show that Pv ⩽ q, so suppose that (1 − q)Pv = 0. The central support of 1 ⊗ eB2 in A1 ⊗ M2,eB2 is 1, so there is a nonzero subprojection Q of (1 − q)Pv in A1 ⊗ M2,eB2 with Q 1 ⊗ eB2 in this algebra. The projection Pv has the form Pv = n⩾0 v∗ neBvn where n⩾0 v∗ nvn is the central support z ∈ Z(B) of v∗v ∈ B. With this notation, (3.20) becomes N(Pv) = {x ∈ zMz: xPv = Pvx}. By Lemma 3.8(iii), there is a projection f ∈ N(Pv) so that Q = f Pv. Both Q and Pv commute with B1 ⊗ 1, so the relation (b1 ⊗ 1)Q − Q(b1 ⊗ 1) = 0 for b1 ∈ B1 becomes ((b1 ⊗ 1)f − f (b1 ⊗ 1))Pv = 0. The element (b1 ⊗ 1)f − f (b1 ⊗ 1) lies in N(Pv) so, by Lemma 3.8(i), (b1 ⊗1)f = f (b1 ⊗1) for all b1 ∈ B1. This shows that f ∈ B 1 ∩N(Pv). Moreover, f = (1−q)f follows from the equation 0 = qQ = qf Pv, (4.24) which implies that qf = 0 since qf ∈ N(Pv). Note that (1 − q)z = 0, otherwise zf = 0. Now B 1 ∩ N(Pv) ⊆ B 1 ∩ M = Z(B1) ⊗ M2 (4.25) and Z(B1) ⊗ B2 ∩ Z(B1) ⊗ M2 = Z(B1) ⊗ Z(B2) ⊆ Z(B1) ⊗ B2. (4.26) Thus the inclusion (1 − q)z Z(B1) ⊗ B2 z(1 − q) ⊆ (1 − q)z B 1 ∩ N(Pv) z(1 − q) (4.27)
- 46. 42 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 has the property that the first algebra contains its relative commutant in the second algebra, which is the hypothesis of Lemma 2.6. Thus we may choose a nonzero projection b ∈ (1−q)z(Z(B1)⊗ B2)z(1 − q) with b f in (1 − q)z(B 1 ∩ N(Pv))z(1 − q). Let w be a partial isometry in this algebra with w∗w = b and ww∗ ⩽ f , and note that w commutes with Pv by definition of N(Pv). Then bPv = w∗ wPv = Pvw∗ wPv ∼ wPvw∗ = ww∗ Pv ⩽ f Pv (4.28) in A1 ⊗ M2,eB2 . Since b ⩽ z, Lemma 3.8(i) ensures that bPv = 0. Moreover, bPv 1 ⊗ eB2 in A1 ⊗ M2,eB2 since f Pv has this property. Consider now a projection r ∈ (A1 ⊗ Z(B2))b. The inequality Φ ⊗ Tr2(Pvr) ⩽ Φ ⊗ τ(r) (4.29) is valid by Lemma 4.5. Thus the hypotheses of Theorem 4.1 are satisfied with P replaced by bPv. We conclude that there is an element w ∈ GN (1) A1⊗M2 (A1 ⊗ B2) so that bPv = w∗(1 ⊗ eB2 )w. Since b lies under 1 − q, this contradicts maximality of the qi’s, proving that Pvq = Pv. Thus Pv = ∞ n=1 w∗ n(1 ⊗ eB2 )wn, (4.30) which we also write as Pv = ∞ n=1 W∗ n Wn where Wn is defined to be (1 ⊗ eB2 )wn ∈ A1 ⊗ M2,eB2 . As in Lemma 4.5, we regard (A1,Tr1) as L∞(Ω) for a σ-finite measure space (Ω,μ). We can then identify A1 ⊗ M2,eB2 with L∞(Ω, M2,eB2 ), and we write elements of this tensor product as uniformly bounded measurable functions on Ω with values in M2,eB2 . Then Tr(Pv) = ∞ n=1 Tr2 Wn(ω)∗ Wn(ω) dμ(ω). (4.31) Since Tr(Pv) ∞, we may neglect a countable number of null sets to conclude that Tr2 Pv(ω) = Tr2 ∞ n=1 Wn(ω)∗ Wn(ω) ∞, ω ∈ Ω, (4.32) from which it follows that Pv(ω) ⩽ Q2 for ω ∈ Ω. Thus Pv ⩽ 1⊗Q2 which gives Pv ⩽ S1 ⊗Q2, since the inequality Pv ⩽ S1 ⊗ S2 has already been established. 2 We are now in a position to approximate an intertwiner in a tensor product. Theorem 4.7. Let Bi ⊆ Mi, i = 1,2, be inclusions of finite von Neumann algebras satisfying B i ∩ Mi ⊆ Bi. Given v ∈ GN (1) M1⊗M2 (B1 ⊗ B2) and ε 0, there exist x1,...,xk ∈ B1 ⊗ B2, w1,1,...,w1,k ∈ GN (1) M1 (B1) and w2,1,...,w2,k ∈ GN (1) M2 (B2) such that:
- 47. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 43 1. xj ⩽ 1 for each j; 2. v − k j=1 xj (w1,j ⊗ w2,j ) 2,τ ε. (4.33) Proof. Write B ⊆ M for the inclusion B1 ⊗ B2 ⊆ M1 ⊗ M2 and fix v ∈ GN (1) M (B). Recall from Lemmas 3.6 and 4.6 that there is a projection Pv ∈ Z(B ∩ M,eB ) satisfying Pvv∗v = v∗eBv, and Pv ⩽ Q1 ⊗ Q2 where Qi is the supremum of the set Pi of projections in Z(B i ∩ Mi,eBi ) specified in Definition 4.4. Thus Qi = k Ri,k for some countable sum of orthogonal projections Ri,k ∈ Pi. By Lemma 3.8(iii), any subprojection of Ri,k in Z(B i ∩ Mi,eBi ) is also in Pi, so it follows that every subprojection of Qi in B ∩ Mi,eBi is also of the form k R i,k for some countable sum of orthogonal projections R i,k ∈ Pi. The restriction Φi of Tri to Z(B i ∩ Mi,eBi )Qi is a normal semifinite weight on this abelian von Neumann algebra Ai, and (Ai,Tri) can be identified with L∞(Ωi) for a σ-finite mea- sure space (Ωi,Σi,μi). Since Pv ⩽ Q1 ⊗ Q2, this operator can be viewed as an element of L∞(Ω1 × Ω2,μ1 × μ2), and it also lies in the corresponding L2-space since Tr(Pv) ⩽ 1. By Lemma 2.7 and the previous paragraph, Pv can be approximated in · 2,Tr-norm by finite sums of orthogonal projections of the form R1 ⊗ R2, each lying in Pi. These elementary ten- sors correspond to measurable rectangles in Ω1 ×Ω2. By the definition of Pi, each Ri is close in · 2,Tri -norm to a finite sum k j=1 w∗ i,j eBi wi,j , with wi,j ∈ GN (1) Mi (Bi). This allows us to make the following approximation: given ε 0, there exist finite sets {wi,j }k j=1 ∈ GN (1) Mi (Bi), i = 1,2, such that Pv − k j=1 w∗ 1,j eB1 w1,j ⊗ w∗ 2,j eB2 w2,j 2,Tr ε. (4.34) If we multiply on the right in (4.34) by v∗eB = v∗eBvv∗eB, then the result is v∗ eB − k j=1 w∗ 1,j eB1 w1,j ⊗ w∗ 2,j eB2 w2,j v∗ eB 2,Tr ε, (4.35) using the fact that Pvv∗v = v∗eBv. A typical element of the sum in (4.35) is w∗ 1j ⊗ w∗ 2,j (eB1 ⊗ eB2 )(w1,j ⊗ w2,j )v∗ (eB1 ⊗ eB2 ) which has the form (w∗ 1,j ⊗ w∗ 2,j )x∗ j eB, where x∗ j = EB((w1,j ⊗ w2,j )v∗) ∈ B has x∗ j ⩽ (w1,j ⊗ w2,j )v∗ ⩽ 1. Thus (4.35) becomes v∗ − k j=1 (w1,j ⊗ w2,j )∗ x∗ j eB 2,Tr ε. (4.36) For each y ∈ M, yeB 2 2,Tr = Tr eBy∗ yeB = τ y∗ y = y 2 2,τ , (4.37)
- 48. 44 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 and so (4.36) implies that v − k j=1 xj (w1,j ⊗ w2,j ) 2,τ ε, as required. 2 Corollary 4.8. Let Bi ⊆ Mi, i = 1,2, be inclusions of finite von Neumann algebras satisfying B i ∩ Mi ⊆ Bi. Then W∗ GN(1) (B1 ⊗ B2) = W∗ GN(1) (B1) ⊗ W∗ GN(1) (B2) . (4.38) There are two extreme cases where the hypothesis B ∩ M ⊆ B is satisfied. The first is when B is an irreducible subfactor where the result of Theorem 4.7 can be deduced from the stronger results of [20]. The second is when B is a masa in M where Theorem 4.7 is already known [1]. The following example explains the preference given to intertwiners over unitary normalizers in intermediate cases, even in a simple setting. Example 4.9. Let M be a II1 factor, let p ∈ M be projection whose trace lies in (0,1/2), and let B = pMp + (1 − p)M(1 − p). This subalgebra has no non-trivial unitary normaliz- ers, essentially because τ(p) = τ(1 − p). However, the tensor product B ⊗ B ⊆ M ⊗ M does have such normalizers because the compressions by p ⊗ (1 − p) and (1 − p) ⊗ p, which have equal traces, are conjugate by a unitary normalizer u which is certainly outside B ⊗ B. Accord- ing to Theorem 4.7, u can be obtained as the limit of finite sums of elementary tensors from W∗(GN(1) (B)) ⊗ W∗(GN(1) (B)). 5. Groupoid normalizers of tensor products In this section we return to the groupoid normalizers GNM(B), namely those v ∈ M such that v,v∗ ∈ GN (1) M (B). Our objective in this section is to establish a corresponding version of Theorem 4.7 for GN(B), and consequently we will assume throughout that any inclusion B ⊆ M satisfies the relative commutant condition B ∩ M ⊆ B. We will need to draw a sharp distinction between those intertwiners v that are groupoid nor- malizers and those that are not, and so we introduce the following definition. Definition 5.1. Say that v ∈ GN(1) (B) is strictly one-sided if the only projection p ∈ Z(Bv∗v) = Z(B)v∗v for which vp ∈ GN(B) is p = 0. When B is an irreducible subfactor of M then any unitary u ∈ M satisfying uBu∗ B is a strictly one-sided intertwiner (see [20, Example 5.4] for examples of such unitaries). Given v ∈ GN(B), recall from Section 3 that there is a projection Pv ∈ Z(B ∩ M,eB ) such that Pvv∗v = v∗eBv, and Pv has the form Pv = n⩾0 v∗ neBvn, (5.1)
- 49. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 45 where there exist partial isometries wn ∈ B so that vn = vw∗ n ∈ GN(1) (B), and w∗ mwn = 0 for m = n. Letting pn denote the projection wnw∗ n ∈ B, it also holds that v∗ nvn = pn. We will employ this notation below. Lemma 5.2. Let v ∈ GN(B), and let u ∈ GN(1) (B) be strictly one-sided. Then Pvu∗eBu = 0. Proof. By Lemmas 3.2(i) and 3.6, u∗eBu ∈ Z(B ∩ M,eB )u∗u and Pv ∈ Z(B ∩ M,eB ), showing that Pv and u∗eBu are commuting projections. Let Q denote the projection Pvu∗eBu in Z(B ∩ M,eB )u∗u which lies below both Pv and u∗eBu. From Lemmas 3.4 and 3.8 we may find projections z ∈ Z(B) and f ∈ M ∩ {Pv} such that Q = zu∗ eBu = f Pv. (5.2) From (5.1), write Pv as the strongly convergent sum Pv = n⩾0 wnv∗ eBvw∗ n, (5.3) so that (5.2) becomes n⩾0 f wnv∗ eBvw∗ n = zu∗ eBu. (5.4) If we multiply (5.4) on the right by pj = wj w∗ j and on the left by eBu, noting that uzu∗ ∈ B, then the result is eBbj vw∗ j = eBuzu∗ upj = eBuu∗ uzpj = eBuzpj (5.5) for each j ⩾ 0, where bj = EB(uf wj v∗) ∈ B. Thus bj vw∗ j = uzpj , j ⩾ 0. (5.6) If we sum (5.6) over j ⩾ 0, then the right-hand side will converge strongly, implying strong convergence of j⩾0 bj vw∗ j . If we return to (5.4) and multiply on the left by eBu, then we obtain uz = n⩾0 bnvw∗ n (5.7) with strong convergence of this sum. It follows that, for each b ∈ B, zu∗ buz = lim k→∞ m,n⩽k wnv∗ b∗ nbbmvw∗ m (5.8) strongly, and thus zu∗buz ∈ B since v∗b∗ nbbmv ∈ B. Thus the projection p = zu∗u ∈ Z(Bu∗u) satisfies pu∗Bup ⊆ B. Since u is strictly one-sided, we conclude that zu∗u = 0, showing that
- 50. 46 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 Q = zu∗ eBu = 0 (5.9) from (5.2). This proves the result. 2 We can use the preceding lemma to show that a strictly one-sided intertwiner v ∈ GN(1) (B) has the property that the only projection p ∈ Bv∗v for which vp ∈ GN(B) is p = 0. Indeed, take such a projection p for which w = vp ∈ GN(B). Then Pvw∗w = w∗eBw so that Pv ⩾ Pw by Remark 3.7. Lemma 5.2 then gives Pwv∗eBv = 0. Thus Pww∗eBw = w∗eBw = 0 and so w = 0. Lemma 5.3. Let v ∈ GN(1) (B). Then there exist orthogonal families of orthogonal projections en,fn ∈ v∗vBv∗v such that (en + fn) = v∗v, each ven is strictly one-sided, and each vfn lies in GN(B). Proof. Let {en} be a maximal orthogonal family of projections in Z(B)v∗v such that ven is strictly one-sided, and set e = en. Then choose a maximal family of orthogonal projections {fn} ∈ Z(B)(v∗v − e) such that vfn ∈ GN(B). If en + fn = v∗v then the result is proved, so consider the projection g = v∗v − en − fn ∈ Z(B)v∗v and suppose that g = 0. Then vg cannot be strictly one-sided otherwise the maximality of {en} would be contradicted. Thus there exists z ∈ Z(B) such that vgz is a nonzero element of GN(B). But this contradicts maximality of {fn}, proving the result. 2 We now return to considering two containments Bi ⊆ Mi satisfying B i ∩ Mi ⊆ Bi, and the tensor product containment B = B1 ⊗ B2 ⊆ M1 ⊗ M2 = M. The next lemma is the key step required to obtain a version of Theorem 4.7 for groupoid normalizers. We need a result from the perturbation theory of finite von Neumann algebras. For any containment A ⊆ N, where N has a specified trace τ, recall that N ⊂δ,τ A means that sup x − EA(x) 2 : x ∈ N, x ⩽ 1 ⩽ δ where · 2 is defined using the given trace τ. If τ is scaled by a constant λ, then ⊂δ,λτ is the same as ⊂δ/ √ λ,τ . Then [16, Theorem 3.5] (see also [18, Theorem 10.3.5]), stated for normalized traces, has the following general interpretation: if A ⊆ N and N ⊂δ,τ A for some δ (τ(1)/23)1/2, then there exists a nonzero projection p ∈ Z(A ∩ N) such that Ap = pNp. Lemma 5.4. Let v ∈ GN(1) (B1) and w ∈ GN(1) (B2). If v (or w) is strictly one-sided then v ⊗ w ∈ GN(1) (B) is strictly one-sided. Proof. Without loss of generality, suppose that v is strictly one-sided. Fix a nonzero projection p ∈ Z(B)(v∗v ⊗ w∗w), and let τi be the faithful normalized normal trace on Mi ⊃ Bi. Given ε 0, we may choose projections pi ∈ Z(B1)v∗v and qi ∈ Z(B2)w∗w, 1 ⩽ i ⩽ k, with p − k i=1 pi ⊗qi 2,τ ε and the pi’s orthogonal since these projections lie in abelian von Neumann algebras. Here, · 2,τ is with respect to the normalized trace τ = τ1 ⊗ τ2 on M. Now piB1pi = piv∗ vB1v∗ vpi ⊆ piv∗ B1vpi (5.10) since vB1v∗ ⊆ B1. If it were true that piv∗B1vpi ⊂δ,τ1 piB1pi for some δ (τ(pi)/23)1/2, then it would follow from [16, Theorem 3.5] that there exists a nonzero projection p i ∈ (piB1pi) ∩
- 51. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 47 piv∗B1vpi ⊆ Z(B1)pi such that p ipiB1pi = p ipiv∗B1vpip i, contradicting the hypothesis that v is strictly one-sided. Thus there exists bi ∈ B1 satisfying piv∗ bivpi ⩽ 1, d2,τ1 piv∗ bivpi,piB1pi ⩾ τ1(pi)/23 1/2 , (5.11) where d2,τ1 (x,A) = inf{ x − a 2,τ1 : a ∈ A} for any von Neumann algebra A. Since each pi lies under v∗v, the projections vpiv∗ lie in B1 and are orthogonal. We may then define an element b ∈ B by b = i vpiv∗ bivpiv∗ ⊗ wqiw∗ , (5.12) and the orthogonality gives b ⩽ 1. Moreover, (v ⊗ w)∗ b(v ⊗ w) = i piv∗ bivpi ⊗ qi, (5.13) and d2,τ (v ⊗ w)∗ b(v ⊗ w),B 2 ⩾ i τ1(pi)τ2(qi) /23 τ(p) − ε /23. (5.14) The right-hand side of (5.13) is unchanged by pre- and post-multiplication by the projection i pi ⊗ qi, and this is close to p. This leads to the estimate p(v ⊗ w)∗ b(v ⊗ w)p − (v ⊗ w)∗ b(v ⊗ w) 2,τ ⩽ 2 p − i pi ⊗ qi 2,τ 2ε. (5.15) From (5.14) and (5.15), d2,τ p(v ⊗ w)∗ b(v ⊗ w)p,B τ(p) − ε /23 1/2 − 2ε. (5.16) A sufficiently small choice of ε then shows that p(v ⊗ w)∗b(v ⊗ w)p / ∈ B, and thus v ⊗ w is strictly one-sided. 2 We can now give the two-sided counterpart of Theorem 4.7. Theorem 5.5. Let Bi ⊆ Mi, i = 1,2, be inclusions of finite von Neumann algebras satisfying B i ∩ Mi ⊆ Bi. Given v ∈ GNM1⊗M2 (B1 ⊗ B2) and ε 0, there exist x1,...,xk ∈ B1 ⊗ B2, w1,1,...,w1,k ∈ GNM1 (B1) and w2,1,...,w2,k ∈ GNM2 (B2) such that: 1. xj ⩽ 1 for each j; 2. v − k j=1 xj (w1,j ⊗ w2,j ) 2,τ ε. (5.17)
- 52. 48 J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 Proof. Consider v ∈ GNM1⊗M2 (B1 ⊗ B2). Following the proof of Theorem 4.7, given ε 0, there exist elements vi,j ∈ GN(1) (Bi), 1 ⩽ j ⩽ k, so that Pv − k j=1 (v1,j ⊗ v2,j )∗ eB(v1,j ⊗ v2,j ) 2,Tr ε (5.18) as in (4.34). Using Lemma 5.3, we may replace this sum with one of the form (w1,j ⊗ w2,j )∗ eB(w1,j ⊗ w2,j ) + (x1,j ⊗ x2,j )∗ eB(x1,j ⊗ x2,j ) where the wi,j ’s are two-sided and, for each j, at least one of x1,j ,x2,j is strictly one-sided. By Lemma 5.4, each x1,j ⊗ x2,j is strictly one-sided, so (x1,j ⊗ x2,j )∗eB(x1,j ⊗ x2,j )Pv = 0 by Lemma 5.2. If we multiply on the right by Pv, then Pv − j (w1,j ⊗ w2,j )∗ eB(w1,j ⊗ w2,j )Pv 2,τ ε. (5.19) Simple approximation allows us to obtain the same estimate for some finite subcollection of the w1,j and w2,j , say w1,1,...,w1,k and w2,1,...,w2,k. We now continue to follow the proof of Theorem 4.7 from (4.34) to obtain the required xj . 2 Just as in Section 4, the next corollary follows immediately. Corollary 5.6. Let Bi ⊆ Mi, i = 1,2, be inclusions of finite von Neumann algebras satisfying B i ∩ Mi ⊆ Bi. Then GNM1 (B1) ⊗ GNM2 (B2) = GNM1⊗M2 (B1 ⊗ B2) . Acknowledgments The work in this paper originated during the Workshop in Analysis and Probability, held at Texas AM University during Summer 2007. It is a pleasure to express our thanks to both the organizers of the workshop and to the NSF for providing financial support to the workshop. References [1] I. Chifan, On the normalizing algebra of a masa in a II1 factor, arXiv:math.OA/0606225, 2006. [2] E. Christensen, Subalgebras of a finite algebra, Math. Ann. 243 (1979) 17–29. [3] J. Dixmier, Sous-anneaux abéliens maximaux dans les facteurs de type fini, Ann. of Math. (2) 59 (1954) 279–286. [4] H.A. Dye, On groups of measure preserving transformation, I, Amer. J. Math. 81 (1959) 119–159. [5] E.G. Effros, Z.-J. Ruan, On approximation properties for operator spaces, Internat. J. Math. 1 (2) (1990) 163–187. [6] J. Fang, On completely singular von Neumann subalgebras, Proc. Edinb. Math. Soc. (2) 52 (2009) 607–618. [7] J. Feldman, C.C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, I, Trans. Amer. Math. Soc. 234 (2) (1977) 289–324. [8] J. Feldman, C.C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, II, Trans. Amer. Math. Soc. 234 (2) (1977) 325–359. [9] V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1) (1983) 1–25.
- 53. J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49 49 [10] V. Jones, V.S. Sunder, Introduction to Subfactors, London Math. Soc. Lecture Note Ser., vol. 234, Cambridge University Press, Cambridge, 1997. [11] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. I, Grad. Stud. Math., vol. 15, American Mathematical Society, Providence, RI, 1997, Elementary theory, reprint of the 1983 original. [12] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. II, Grad. Stud. Math., vol. 16, American Mathematical Society, Providence, RI, 1997, Advanced theory, corrected reprint of the 1986 original. [13] M. Pimsner, S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1) (1986) 57–106. [14] S. Popa, On a class of type II1 factors with Betti numbers invariants, Ann. of Math. (2) 163 (3) (2006) 809–899. [15] S. Popa, Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I, Invent. Math. 165 (2006) 369–408. [16] S. Popa, A.M. Sinclair, R.R. Smith, Perturbations of subalgebras of type II1 factors, J. Funct. Anal. 213 (2) (2004) 346–379. [17] H.L. Royden, Real Analysis, third ed., Macmillan Publishing Company, New York, 1988. [18] A.M. Sinclair, R.R. Smith, Finite von Neumann Algebras and Masas, London Math. Soc. Lecture Note Ser., vol. 351, Cambridge University Press, Cambridge, 2008. [19] A.M. Sinclair, R.R. Smith, S.A. White, A. Wiggins, Strong singularity of singular masas in II1 factors, Illinois J. Math. 51 (4) (2007) 1077–1083. [20] R.R. Smith, S.A. White, A. Wiggins, Normalizers of irreducible subfactors, J. Math. Anal. Appl. 352 (2) (2009) 684–695. [21] M. Takesaki, Theory of Operator Algebras, II, Operator Algebras and Non-commutative Geometry, vol. VI, Ency- clopaedia Math. Sci., vol. 125, Springer-Verlag, Berlin, 2003.
- 54. Journal of Functional Analysis 258 (2010) 50–66 www.elsevier.com/locate/jfa Linear maps preserving the minimum and reduced minimum moduli A. Bourhim a,∗,1,2 , M. Burgos b,3 , V.S. Shulman c a Département de Mathématiques et de Statistique, Université Laval, Québec (Québec), Canada G1K 7P4 b Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain c Department of Mathematics, Vologda State Technical University, Vologda, Russia Received 16 September 2008; accepted 2 October 2009 Communicated by Paul Malliavin Abstract We describe linear maps from a C∗-algebra onto another one preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the reduced minimum modulus. © 2009 Elsevier Inc. All rights reserved. Keywords: C∗-algebras; Linear preserver problems; Minimum modulus; Surjectivity modulus; Reduced minimum modulus 1. Introduction In the last few decades, there has been a considerable interest in the so-called linear preserver problems which concern the characterization of linear or additive maps on matrix algebras or operator algebras or more generally on Banach algebras that leave invariant a certain function, a certain subset, or a certain relation; see for instance the survey papers [8,13,21,22,29] and the * Corresponding author. E-mail addresses: bourhim@mat.ulaval.ca, abourhim@syr.edu (A. Bourhim), mariaburgos@ugr.es (M. Burgos), shulman_v@yahoo.com (V.S. Shulman). 1 Current address: Department of Mathematics, Syracuse University, 215 Carnegie Building, Syracuse, NY 13244, USA. 2 Supported by an adjunct professorship at Laval university. 3 Partially supported by the Junta de Andalucía PAI project FQM-3737 and the I+D MEC project MTM2007-65959. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.003
- 55. Other documents randomly have different content
- 56. although it was observed about this time that six loads of money reached Valladolid for him. Charles V, from his retirement of Yuste, wrote to him, May 18th, expressing surprise that he, the creature of imperial favor, should hesitate to repay the benefits conferred, especially as he could have what security he desired for the loan. This letter, with one from Juana, was conveyed to him by Hernando de Ochoa, whose report to Charles, May 28th, of the interview, showed how little respect was felt for the man. Ochoa reproached him with having promised to see what he could do, in place of which he had gone into hiding at San Martin de la Fuente, fourteen leagues from the court at Valladolid, where he had lain for two months, hoping that the matter would blow over. “He said to me, before a consecrated host, that the devils could fly away with him if ever he had 100,000 or 80,000, or 60,000, or 30,000 ducats, for he had always spent much in charities and had made dotations amounting to 150,000.” Ochoa pressed him hard; he admitted that his archbishopric, which he had held since 1546, was worth 60,000 ducats a year and Ochoa showed that, admitting his claims for charities and expenses, he had laid aside at least 30,000 a year “which you cannot possibly have spent, for you never have any one to dine in your house and you do not accumulate silver plate, like other gentlemen; all this is notorious, and the whole court knows it.... This embarrassed him, but he repeated with great oaths that he had no money, that it was not well thus to oppress prelates, nor would money thus obtained be lucky for war; God would help the king and what would Christendom say about it.” The honest Ochoa still urged him to return to the court and save his honor, intimating that the king might take action that would be highly unpleasant, but it was to no purpose. Valdés was obdurate and clung resolutely to his shekels.[133] Philip had sent instructions as to the treatment of recalcitrants—probably relegating bishops to their sees and nobles to their estates—but there was hesitation felt as to banishing Valdés from the court, although the continued pressure of Charles and Juana only extorted a promise of fifty thousand ducats. Yet it was desired to remove him and plans were tried to offer him a pretext for going. In March, 1588, Juana ordered him to accompany the body of Queen Juana la loca to Granada for interment, from which place he could visit his Seville church; he made excuses but promised to go shortly. Then, when she repeated the order, he offered many reasons for evading it, including the heresies recently discovered in Seville and Murcia; the
- 57. translation of the body could wait until September and everybody, he said, was trying to drive him from the court. She referred the matter to the Royal Council, which decided that his excuses were insufficient and that, even if the interment were postponed he could properly be ordered to reside in his see.[134] It was evident to Valdés that something was necessary to strengthen his position and he skilfully utilized the discovery of a few Protestants in Valladolid, of whom some were eminent clerics, like Augustin Cazalla and Fray Domingo de Rojas, and others were persons of quality, like Luis de Rojas and Doña Ana Enríquez. We shall have occasion to note hereafter the extraordinary excitement caused by the revelation that Protestantism was making inroads in court circles, the extent of which was readily exaggerated, and it was stimulated and exploited by Valdés, who magnified his zeal in combating the danger and conjured, at least for the moment, the storm that was brewing. Philip wrote from Flanders, June 5, 1558, to send him to his see without delay; if he still made excuses he was to be excluded from the Council of State and this would answer until his approaching return to Spain, when he would take whatever action was necessary. Ten days later, on receiving letters from Valdés enumerating the prisoners and describing the efforts made to avert the danger, he countermanded the orders.[135] Still, this was only a respite; we chance to hear of a meeting of the Council of State, in August or September, in which Juan de Vega characterized as a great scandal the disobedience of a vassal to the royal commands, in a matter so just as residence in his see, and he suggested that, when the court moved, no quarters should be assigned to Valdés, to which Archbishop Carranza replied that it was no wonder that the orders of the king were unable to effect what the commandments of God and the Church could not accomplish.[136] Something further was necessary to render him indispensable— something that could be prolonged indefinitely and if, at the same time, it would afford substantial relief to the treasury, he might be forgiven the niggardness that had resisted the appeals of the sovereign. He had for some time been preparing a scheme for this, which was nothing less than the prosecution of the Primate of the Spanish Church, the income of whose see was rated at from 150,000 to 200,000 ducats. To measure the full audacity of this it is necessary to appreciate the standing of Archbishop Carranza.
- 58. ARCHBISHOP CARRANZA Bartolomé de Carranza y Miranda was born in 1503. At the age of 12 he entered the university of Alcalá; at 18 he took the final vows of the Dominican Order and was sent to study theology in the college of San Gregorio at Valladolid, where, in 1530, he was made professor of arts, in 1533 junior professor of theology and, in 1534, chief professor as well as consultor of the tribunal of Valladolid. In 1540 he was sent as representative of his Order to the General Chapter held in Rome, where he distinguished himself and was honored with the doctorate, while Paul III granted him a licence to read prohibited heretic books. On his return to Spain his reputation was national; he was largely employed by the Suprema in the censorship of books, especially of foreign Bibles, while the Councils of Indies and Castile frequently submitted intricate questions for his judgement. In 1542 he was offered the see of Cuzco, esteemed the wealthiest in the colonies, when he replied that he would willingly go to the Indies on the emperor’s service but not to undertake the cure of souls.[137] On the convocation of the Council of Trent, in 1545, Charles V selected him as one of the delegates and, during his three years’ service there, he earned the reputation throughout Christendom of a profound theologian. When, in 1548, Prince Philip went to join his father in Flanders, they both offered him the position of confessor which he declined, as he did the see of Canaries which was tendered to him in 1550. In this latter year he was elected provincial of his Order for Castile and, in 1551, he was sent to the second convocation of the Council of Trent by Charles and also as the representative of Siliceo, Archbishop of Toledo. As usual, he played a prominent part in the Council and, after its hasty dissolution, he remained there for some time employed in the duty of examining and condemning heretical books. In 1553 he returned to his professorship at Valladolid and when, in 1554, Prince Philip sailed for England to marry Queen Mary and restore the island to the unity of the Church, he took Carranza with him as the fittest instrument for the work.[138] Carranza subsequently boasted that, during his three years’ stay in England, he had burnt, reconciled, or driven from the land thirty thousand heretics and had brought two million souls back to the Church. If we may believe his admiring biographers he was the heart and soul of the Marian persecution and Philip did nothing in religious matters without his advice. When, in September, 1555, Philip rejoined his father in Flanders, he left
- 59. ARCHBISHOP CARRANZA Carranza as Mary’s religious adviser, in which capacity he remained until 1557. Regarded by the heretics as the chief cause of their sufferings he barely escaped from repeated attempts on his life by poison or violence. [139] It is true that English authorities of the period make little mention of him, but the continued confidence of Philip is ample evidence that his persecuting zeal was sufficient to satisfy that exacting monarch. When, in 1557, Carranza rejoined Philip in Flanders he was probably engrossed in the preparation and printing of his large work on the Catechism, of which more hereafter, but he still found time to investigate and impede the clandestine trade of sending heretic books to Spain.[140] That he had completely won Philip’s esteem and confidence was seen when Siliceo of Toledo died, May 1, 1557, and Philip appointed him as successor in the archbishopric. He refused the splendid prize and suggested three men as better fitted for the place. Philip persisted; he was going to a neighboring convent to confess and commune prior to the opening of the campaign and ordered Carranza to obey on his return. When he came back he sent the presentation written in his own hand; Carranza yielded, but on condition that, as the war with the pope would delay the issue of the bulls, the king in the interval could make another selection. This effort to avoid the fatal gift was fruitless. On his return from the campaign, Philip in an autograph letter summoned him to fulfil his promise and made the appointment public. So high was Carranza’s reputation that, when the presentation was laid before the consistory in Rome, on December 6th, it was at once confirmed, without observing the preconization, or the customary inquiry into the fitness of the appointee, or a constitution which prohibited final action on the same day. [141] The elevation of a simple friar to the highest place in the Spanish Church was a blow to numerous ambitions that could scarce fail to arouse hostility. Valdés himself was said to have aspirations for the position and to be bitterly disappointed. Pedro de Castro, Bishop of Cuenca, had also cherished hopes and was eager for revenge. Carranza, moreover, was not popular with the hierarchy. He was that unwelcome character, a reformer within the Church and, while everyone acknowledged the necessity of reform, no one looked with favor on a reformer who assailed his profitable abuses. As far back as 1547, while in attendance on the Council of Trent, Carranza had preached a sermon on
- 60. one of the most crying evils of the time, the non-residence of bishops and beneficiaries, and had embodied his views in a tractate as severe as a Lutheran would have written on this abuse and the kindred one of pluralities, to which possibly the stringent Tridentine provisions on the subject may be attributed.[142] Such an outburst was not calculated to win favor, seeing that the splendor of the curia was largely supported by the prelacies and benefices showered upon its members and that in Spain there was scarce an inquisitor or a fiscal who was not a non-resident beneficiary of some preferment. Carranza had, moreover, a peculiarly dangerous enemy in a brother Dominican, Melchor Cano, perhaps the leading Spanish theologian of the time when Spanish theology was beginning to dominate the Church. Learned, able, keen-witted and not particularly scrupulous, he was in intellect vastly superior to Carranza; there had been early rivalry, when both were professors of theology, and causes of strife in the internal politics of the Order had arisen, so that Cano could scarce view without bitterness the sudden elevation of his brother fraile.[143] His position at the time was somewhat precarious. When, in 1556, Paul IV forced war on Philip II, that pious prince sought the advice of theologians as to the propriety of engaging in hostilities with the Vicegerent of God, and the parecer, or opinion which Cano drew up, was an able state paper that attracted wide attention. He defended uncompromisingly the royal prerogatives, he virtually justified the German revolt when the Centum Gravamina of the Diet of Nürnberg, in 1522, were unredressed and he described the corruption of Rome as a disease of such long standing as to be incurable. [144] This hardy defiance irritated Paul in the highest degree. April 21, 1556 he issued a brief summoning that son of perdition, Melchor Cano, to appear before him within sixty days for trial and sentence, but the brief was suppressed by the Royal Council and Cano was ordered not to leave the kingdom. The Spanish Dominicans rallied to his defence; in the chapter of 1558 he was elected provincial and deputy to the general chapter to be held in Rome, but Paul ordered the election to be annulled and Cano to be deprived of his priorate of San Esteban. Cano complained of lukewarmness in his defence on the part of both Philip and Carranza and it is easy to understand that, feeling keenly the disgrace inflicted on him, he was in a temper to attack any one more fortunate than himself.[145]
- 61. ARCHBISHOP CARRANZA At this inauspicious moment Carranza presented himself as a fair object of attack by all who, from different motives, might desire to assail him. If we may judge from his writings, he must have been impulsive and inconsiderate in his speech, given to uttering extreme views which made an impression and then qualifying them with restrictions that were forgotten. He was earnestly desirous of restoring the Church to its ancient purity and by no means reticent in exposing its weaknesses and corruption. He had been trained at a time before the Tridentine definitions had settled points of faith which, since the twelfth century, had been the subjects of debate in the schools, and even in his maturity the Council of Trent had not yet been clothed with the awful authority subsequently accorded to it, for the inglorious exit of its first two convocations, in 1547 and 1552, gave little promise of what lay in the future. The echo of the fierce Lutheran controversies had scarce penetrated into Spain and comparatively little was there known of the debates which were shaking to its centre the venerable structure of the Church. Carranza’s very labors in condemning heretic books and converting heretics had acquainted him with their doctrines and modes of expression; he was a confused thinker and his impulsive utterances were liable to be construed in a sense which he did not anticipate. As early as 1530 he had been denounced to the Inquisition by Fray Juan de Villamartin as a defender of Erasmus, especially in the matter of confession and the authorship of the Apocalypse and, during his persecuting career in England, he more than once gave opportunity, in his sermons, to unfavorable comment.[146] It was also in evidence that when in Rome, in 1539, he had written to Juan de Valdés in Naples, asking what authors should be studied for understanding Scripture, as he would have to teach that subject, and that Valdés replied in a letter which Carranza circulated among his students in Valladolid—a letter highly heretical in its teachings which Valdés subsequently included in his “One hundred and ten Divine Considerations.”[147] It is true that, in 1539, Juan de Valdés was not reckoned a heretic, but, if the letter was correctly identified with the “Consideration” in question its circulation was highly imprudent, for it asserted that the guides for the study of Scripture are prayer inspired by God and meditation based on spiritual experience, thus discarding tradition for private interpretation, and it further dwelt upon the confidence which the soul should feel in justification through Christ. In the death-struggle with Protestantism the time had passed for easy-going
- 62. latitude of opinion and, in the intricate mazes of scholastic theology, it was necessary to walk warily, for acute censorship could discover heresy in any unguarded expression. The great services rendered by Cardinal Morone and Cardinal Pole did not save them from the prosecuting zeal of Paul IV and Contarini and Sadoleto were both suspect of heresy.[148] Under such conditions a rambling inconsequential thinker like Carranza was peculiarly open to attack. He had unquestionably been more or less intimate with some of the prominent personages whose arrest for Lutheranism, in the spring of 1558, produced so immense a sensation. It was not unnatural that, on their trials, they should seek to shield themselves behind his honored name, but the detached fragments of conversation which were cited in support of vague general assertions, even if correctly reported, amount to nothing in the face of the emphatic testimony by Fray Domingo de Rojas, for the discharge of his conscience, a few hours before his execution, that he had never seen in Carranza anything that was not Catholic in regard to the Roman Church and all its councils, definitions and laws and that when Lutherans were alluded to he said their opinions were crafty and deceiving; they had sprung from hell and the incautious could easily be deceived by them.[149] The credence due to the evidence of the Lutherans, on which so much stress was laid, can be gauged by a subsequent case illustrative of the tendency to render Carranza responsible for all aberrations of belief. A certain Gil Tibobil (de Bonneville) on trial in 1564 for Lutheranism, in Toledo, sought to palliate his guilt by asserting that he had heard Carranza preach, in the church of San Agustin, against candles and images and that confession was to be made to God and not to the priest. This was too crude to be accepted and he was sternly told that it cast doubt on the rest of his confession for, if Carranza had thus preached publicly, it would have come to the knowledge of the Inquisition and he would have been punished.[150] Whether the testimony acquired in the trials of the Lutherans was important or not, Inquisitor-general Valdés lost no time in using it to discredit Carranza in the opinion of the sovereigns. As early as May 12, 1588, in a report to Charles V at Yuste, his assistance is asked in obtaining the arrest of a fugitive, whose capture would be exceedingly important; he had been traced to Castro de Urdiales, where he was to embark for Flanders to find refuge with Carranza or with his companion Fray Juan de
- 63. ARCHBISHOP CARRANZA Villagarcia, where he was sure of being well received. That the real motive was to injure Carranza with Charles appears from Valdés repeating the story to him in a report of June 2, adding that the fugitive had escaped and that information had been sent to Philip in order that he might be captured.[151] It is reasonable to assume that whatever incriminating evidence could be obtained from the prisoners was promptly brought to the notice of the sovereigns and that inferences were unscrupulously asserted as facts. At this critical juncture, Carranza delivered himself into the hands of his enemies. In England and Flanders he had employed the intervals of persecution in composing a work which should set forth the irrefragable truths of the Catholic faith and guard the people from the insidious poison of heretical doctrine. This was a task for which, at such a time, he was peculiarly unfitted. He was not only a loose thinker but a looser writer, diffuse, rambling and discursive, setting down whatever idea chanced to occur to him and wandering off to whatever subjects the idea might suggest. Moreover he was earnest as a reformer within the Church, realizing abuses and exposing them fearlessly—in fact, he declared in the Prologue that his object was to restore the purity and soundness of the primitive Church, which was precisely what the heretics professed as their aim and precisely what the ruling hierarchy most dreaded. [152] Worst of all, he did this in the vulgar tongue, unmindful of the extreme reserve which sought to keep from the people all knowledge of the errors and arguments of the heretics and of the contrast between apostolic simplicity and the splendid sacerdotalism of a wealthy and worldly establishment.[153] This he cast into the form of Commentaries on the Catechism, occupying a folio of nine hundred pages, full of impulsive assertions which, taken by themselves, were of dangerous import, but which were qualified or limited, or contradicted in the next sentence, or the next page, or, perhaps, in the following section. No one, I think, can dispassionately examine the Commentaries without reaching the conviction that Carranza was a sincere and zealous Catholic, however reckless may seem many of his isolated utterances. Nor was his orthodoxy merely academic. He belonged to the Church Militant and his hatred of heresy and heretics breaks out continually, in season and out of season, whether apposite or not to his immediate subject. Heretic arguments are not worthy of confutation—it is enough to say that a doctrine is
- 64. condemned by the Church and therefore it is heretical. The first duty of the king is to preserve his dominions in the true faith and to chastise those who sin against it. Even if heretics should perform miracles, their disorderly lives and corrupted morals would be sufficient to guard the people from listening to them or believing them. If they do not admit their errors they are to be condemned to death; this is the best theology that a Christian can learn and it was not more necessary in the time of Moses than it is at present.[154] Even in that age, when theology was so favorite a topic, few could be expected to wade through so enormous a mass of confused thinking and disjointed writing, and it was easy for Carranza’s enemies to garble isolated sentences by which he could be represented to the sovereigns as being at least suspect in the faith, and suspicion of heresy was quite sufficient to require prosecution. Carranza himself, after his book was printed, seems to have felt apprehension and to have proceeded cautiously in giving it to the public. A set of the sheets was sent to the Marchioness of Alcañizes and a dozen or more copies were allowed to reach Spain, where they were received in March, 1558. Pedro de Castro, Bishop of Cuenca, obtained one and speedily wrote to Valdés, denouncing the writer as guilty of heretical opinions. Valdés grasped the opportunity and ordered Melchor Cano to examine the work. Cano took as a colleague Fray Domingo de Cuevas and had no difficulty in discovering a hundred and one passages of heretical import. The preliminaries to a formal trial were now fairly under way, the result of which could scarce be doubtful under inquisitorial methods, if the royal and papal assent could be obtained, necessary even to the Inquisition before it could openly attack the Primate of the Spanish Church. Despite the profound secrecy enveloping the operations of the Inquisition, it was impossible that, in an affair of such moment, there should not be indiscretions and Carranza in Flanders was advised of what was on foot. His friends urged him not to return to Spain but to take refuge in Rome under papal protection, but he knew that this would irrevocably cost him the favor of Philip, for exaggerated jealousy of papal interference with the Inquisition was traditional since the time of Ferdinand and Isabella, and he virtually surrendered his case at once by instructing his printer, Martin Nucio, not to sell copies of the Commentaries without his express orders, thus withdrawing it from circulation.[155]
- 65. ARCHBISHOP CARRANZA But little adverse impression seems as yet to have been made on Philip. When Carranza was about to leave Flanders, the king gave him detailed instructions which manifest unbounded confidence. He was to go directly to Valladolid and represent the extreme need of money; then he was to see Queen Mary of Hungary, Charles’ sister, and persuade her to come to Flanders; then he was to hasten to Yuste where Philip, through him, unbosomed himself to his father, revealing all his necessities and desires in family as well as in state affairs. In short, Carranza was still one whom he could safely entrust with his most secret thoughts.[156] Carranza, with his customary lack of worldly wisdom, threw away all the advantages of his position. Landing at Laredo on August 1st, he passed through Burgos, where he was involved in an unseemly squabble with the archbishop over his assumed right to carry his archiepiscopal cross in public. He did not reach Valladolid until the 13th and there he tarried, busied ostensibly with a suit between his see and the Marquis of Camarasa over the valuable Adelantamiento of Cazorla, but doubtless occupied also with efforts to counteract the intrigues of Valdés. Then he performed his mission to Mary of Hungary and it was not until the middle of September that he set out on a leisurely journey to Yuste. Valdés had taken care to forestall his visit. An autograph letter of the Princess Juana to Charles, August 8th, says that Valdés had asked her to warn him to be cautious in dealing with Carranza, for he had been implicated by the Lutheran prisoners and would already have been arrested had he been anyone else. Charles was naturally impatient to see him, not only to obtain explanations as to this, but also to receive the messages expected from Philip, for which he was waiting before writing to Flanders. Carranza’s delay, in spite of repeated urgency from Yuste, could not but create a sinister impression and all chance of justification was lost, for Charles was prostrated by his fatal illness before Carranza left Valladolid and the end was near when he reached Yuste about noon on September 20th. Charles expired the next morning at half-past two, Carranza administering to him the last consolations, his method in which formed one of the charges against him on his trial. He had thrown away his last chance and the unexpected death of Charles deprived him of one who might possibly have stood between him and his fate.[157]
- 66. ARCHBISHOP CARRANZA The plans of Valdés were now sufficiently advanced for him to seek the papal authorization which alone was lacking, and his method to obtain this was characteristically insidious. The Suprema addressed, September 9th, to Paul IV a relation of its labors in discovering and prosecuting the Lutheran heretics. There was skilful exaggeration of the danger impending from a movement, the extent of which could not be known, and it was pointed out that sympathy with the sectaries might be entertained by officials of the Inquisition itself, by the Ordinaries and the consultors; so that extraordinary powers were asked to arrest and judge and relax those suspected or guilty, even though they were persons holding a secular or pontifical and ecclesiastical dignity or belonging to any religious or other Order.[158] As the Inquisition already had jurisdiction over all but bishops (it had not hesitated to arrest and try the Dominican Fray Domingo de Rojas) the self- evident object of this was to obtain surreptitiously, under cover of the word “pontifical,” some general expression that might be used to deprive Carranza of his right to trial by the pope. The Dean of Oviedo, a nephew of Valdés, was sent to Rome as a special agent to procure the desired brief; whether royal sanction for this application was obtained does not appear, but it probably was not, at least at this stage. Carranza meanwhile had been vainly endeavoring to get copies of the censures on his book in order to answer them. He appealed earnestly to his friends in Philip’s court and in Rome but, without awaiting their replies, he pursued his policy of submission and, on September 21st, the day of Charles’s death, he wrote to Sancho López de Otálora, a member of the Suprema, that he consented to the prohibition of his work, provided this was confined to Spain and that his name was not mentioned.[159] In this and what followed he has been accused of weakness, but it is difficult to see what other course lay open to him. He doubtless still considered his episcopal consecration a guarantee for his personal safety, while his reputation for orthodoxy could best be conserved by not entering into a fruitless contest with a power irresistible in its chosen field of action—a contest, moreover, which would have cost him the royal favor that was his main reliance. In pursuance of this policy he even descended to attempting to propitiate Melchor Cano by offering to do whatever he would recommend. Cano subsequently asserted, with customary mendacity, that Carranza would have
- 67. averted his fate had he adopted any of the means which he devised and advised to save him, but it is difficult to imagine what more he could have done.[160] Towards the close of November he wrote to Valdés and the Suprema and to other influential persons professing his submission. He explained the reasons which had led him to write his book in the vernacular after commencing it in Latin; it could readily be suppressed for, on reaching Valladolid, he had withdrawn the edition from the printer; there were no copies in the bookshops and what he had brought with him he would surrender, while the dozen or so that had been sent to Spain could easily be called in as the recipients were all known. Then, on December 9th, he proposed to the Suprema that the book should be prohibited in Spanish and be returned to him for correction and translation into Latin.[161] Had the real object of Valdés been the ostensible one of preserving the faith, this would have amply sufficed; the book would have been suppressed and the public humiliation of the Archbishop of Toledo, so distinguished for his services to religion, would have been an amply deterrent warning to all indiscreet theologians. It was a not unnatural burst of indignation when, in a letter to Domingo de Soto, November 14th, he bitterly pointed out how the heretics would rejoice to know that Fray Bartolomé de Miranda was treated in Spain as he had treated them in England and Flanders and that, after he had burnt them to enforce the doctrines of his book, it was pronounced in Spain unfit to be read.[162] Carranza’s submission brought no result save to encourage his enemies, who put him off with vague replies while awaiting the success of their application to the pope. Meanwhile he had reached Toledo, October 13th, and had applied himself actively to his duties. He was rigid in the performance of divine service, he visited prisons, hospitals and convents, he put an end to the sale of offices and charging fees for licences, he revised the fee-bill of his court, he enforced the residence of parish priests and was especially careful in the distribution of preferment—in short he was a practical as well as theoretical reformer. His charity also was boundless, for he used to say that all he needed was a Dominican habit and that whatever God gave him was for the poor. Thus during his ten months of incumbency, he distributed more than eighty thousand ducats in marrying orphans, redeeming captives, supporting widows, sending students to universities and in gifts to hospitals. [163] He was a model bishop, and the resolute fidelity with which the
- 68. ARCHBISHOP CARRANZA chapter of Toledo supported his cause to the end shows the impression made on a body which, in Spanish churches, was usually at odds with its prelate. He had likewise not been idle in obtaining favorable opinions of his book from theologians of distinction. In view of the rumors of inquisitorial action, there was risk in praising it, yet nearly all those prominent in Spanish theology bore testimony in its favor. The general view accorded virtually with that of Pedro Guerrero, Archbishop of Granada, than whom no one in the Spanish hierarchy stood higher for learning and piety. The book, he said, was without error and, being in Castilian, was especially useful for parish priests unfamiliar with Latin, wherefore it should be extensively circulated. It was true that there were occasional expressions which, taken by themselves, might on their face seem to be erroneous, but elsewhere it was seen that they must be construed in a Catholic sense. To this effect recorded themselves Domingo and Pedro de Soto, men of the highest reputation, Garrionero Bishop of Almería, Blanco of Orense, Cuesta of Leon, Delgado of Lugo and numerous others.[164] If some of these men belied themselves subsequently and aided in giving the finishing blow to their persecuted brother, we can estimate the pressure brought to bear on them. Valdés speedily utilized the power of the Inquisition to check these appreciations of the Commentaries. When, at the University of Alcalá, the rector, the chancellor, and twenty-two doctors united in declaring the work to be without error or suspicion of error, save that some incautious expressions, disconnected from the context, might be mistaken by hasty readers, Valdés muzzled it and all other learned bodies and individuals by a letter saying that it had come to his notice that learned men of the university had been examining books and giving their opinions. As this produced confusion and contradiction respecting the Index which the Inquisition was preparing, all persons, colleges and universities were forbidden to censure or give an opinion concerning any book without first submitting it to the Suprema, and this under pain of excommunication and a fine of two hundred ducats on each and every one concerned.[165] It was impossible to contend with an adversary armed with such weapons. Not content with this, the rector of the university, Diego Sobaños, was prosecuted by the tribunal of Valladolid for
- 69. the part he had taken in the matter; he was reprimanded, fined and absolved ad cautelam. Similar action was taken against the more prominent of those who had expressed themselves favorably and who, for the most part, were forced to retract.[166] The Inquisition played with loaded dice. Dean Valdés of Oviedo meanwhile had succeeded in his mission to Rome, aided, as Raynaldus assures us, by the express request of Philip, though this is more than doubtful. The brief was dated January 7, 1559; it was addressed to Valdés and recited that, as there were in Spain some prelates suspected of Lutheranism, he was empowered for two years from the receipt of the brief, with the advice of the Suprema, to make investigation and, if sufficient proof were found against any one and there was good reason to apprehend his flight, to arrest and keep him in safe custody, but as soon as possible the pope was to be informed of it and the prisoner was to be sent to him with all the evidence and papers in the case. [167] With the exception of the provision against expected flight, this was merely in accordance with the received practice in the case of bishops, but it was the entering wedge and we shall see how its limitations were disregarded. The brief was received April 8th. In place of complying with it and sending Carranza to Rome with the evidence that had been collecting for nearly a year, a formal trial was secretly commenced. The fiscal presented a clamosa or indictment, on May 6th, asking for Carranza’s arrest and the sequestration of his property, “for having preached, written and dogmatized many errors of Luther.” The evidence was duly laid before calificadores, or censors, who reported accordingly and, on the 13th, there was drawn up a summons to appear and answer to the demand of the fiscal. Before proceeding further, in an affair of such magnitude, it was felt that the assent was required of Philip, who was still in Flanders.[168] As recently as April 4th he had replied encouragingly to an appeal from the persecuted prelate. “I have not wanted to go forward in the matter of your book, about which you wrote to me, until the person whom you were sending should arrive; he has spoken with me today. I had already done something of what is proper in this business. Not to detain the courier who goes with the good news of the conclusion of peace, I do not wish to enlarge in replying to you, but I shall do so shortly and meanwhile I earnestly ask you to make no change in what you have done hitherto and to have recourse to no one but to me, for it
- 70. ARCHBISHOP CARRANZA would be in the highest degree disadvantageous.”[169] Philip evidently thought that only Carranza’s book and not his person was concerned, that the affair was of no great importance and his solicitude was chiefly to prevent any appeal to Rome, a matter in which he fully shared the intense feeling of his predecessors. When Carranza ordered his envoy to Flanders, Fray Hernando de San Ambrosio, to proceed to Rome and secure an approbation of the Commentaries, he replied, April 19th, that all his friends at the court earnestly counselled against; it had been necessary to assure Philip of the falsity of the reports that he had done so, whereupon the king had expressed his satisfaction and had said that any other course would have displeased him.[170] Advantage, for which Carranza foolishly offered the opportunity, was taken of this extreme jealousy to win him over. When the Dominican chapter met, in April, 1559, there was open strife between him and Cano, over a report that Cano had styled him a greater heretic than Luther and that he favored Cazalla and the other prisoners. Carranza demanded his punishment for the slander and sought to defeat his candidacy for the provincialate. In this he failed. Cano’s assertion that he had been misunderstood was accepted; he was again elected provincial and Carranza unwisely carried his complaint to Rome. [171] There it became mixed up with the question of Cano’s confirmation, for Paul IV naturally resented the repeated presentation of that “son of iniquity.” Philip, on the other hand, could not abandon the protection of one whose fault, in papal eyes, was his vindication of the royal prerogative, and he interested himself actively in pressing the confirmation. Paul equivocated and lied and sought some subterfuge which was found in Cano’s consecration, in 1552, as Bishop of Canaries (a post which he had resigned in 1553) which was held to render him ineligible to any position in his Order, and a general decree to that effect was issued in July.[172] All this was skilfully used to prejudice Philip against Carranza. In letters of May 16th to him and of May 22nd and 25th to his confessor Bernardo de Fresneda, Cano with great adroitness and small respect for veracity represented himself as subjected to severe persecution. He had always been Carranza’s friend; he had withheld for seven months his censure of the Commentaries and had yielded only to a threat of excommunication and now Carranza was repaying him by intriguing against the confirmation in
- 71. ARCHBISHOP CARRANZA Rome—the truth being that it was not until the end of June that Carranza’s agent reached there. It was a terrible thing, Cano added, if the archbishop, through his Italian General, could thus wrong him and he could not defend himself. He was resolved to suffer in silence, but the persecution was so bitter that if the king did not speedily come to Spain he would have to seek refuge in Flanders.[173] What, in reality, were his sufferings and what the friendly work on which he was engaged, are indicated by a commission issued to him, May 29th, granting him the extraordinary powers of a substitute inquisitor-general and sending him forth on a roving expedition to gather evidence, compelling everyone whom he might summon to answer whatever questions he might ask.[174] The Suprema and Valdés, moreover, in letters of May 13th and 16th to Philip, adopted the same tone; Cano’s labors throughout the affair had been great and it was hoped that the king would not permit his persecution for the services rendered to God and his majesty; there need be no fear of injustice to Carranza, for the investigation was impartial and dispassionate.[175] Philip had already been informed by Cardinal Pacheco, February 24th and again May 13th, that Carranza had sent to the pope copies of the favorable opinions of his book, asking that it be judged in Rome and that his episcopal privilege of papal jurisdiction be preserved.[176] Whatever intentions he had of befriending Carranza were not proof against the assertions that to his intrigues was attributable the papal interference with Cano’s election. On June 26th he wrote to Cano, expressing his satisfaction and assuring him of his support in Rome and, on the same day, to the Suprema approving its actions as to the Commentaries and expressing his confidence that it would do what was right.[177] In thus authorizing the prosecution he ordered the archbishop’s dignity to be respected and he wrote to the Princess Juana that, to avoid scandal, she should invite him to Valladolid to consult on important matters, so that the trial could proceed without attracting attention.[178] Philip’s letters were received July 10th, but there was still hesitation and it was not until August 3d that the princess wrote, summoning Carranza in haste to Valladolid, where she would have lodgings prepared for him. This she sent, with secret instructions, by the hands of Rodrigo de Castro, a member of the Suprema.[179] Carranza was at Alcalá de Henares, whither Diego Ramírez,
- 72. inquisitor of Toledo, was also despatched, under pretext of publishing the Edict of Faith. Carranza, who suspected a snare, was desirous of postponing his arrival at Valladolid until Philip, on whose protection he still relied, should reach Spain. Accordingly he converted the journey into a visitation, leaving Alcalá on the 16th and passing through Fuente el Saz and Talamanca to Torrelaguna, which he reached on the 20th. On the road he received intimations of what was in store and at Torrelaguna Fray Pedro de Soto came with the news that emissaries had already started to arrest him, which elicited from him a despairing and beseeching letter to Fresneda, the royal confessor.[180] De Soto’s report was true. Valdés dreaded as much as Carranza desired Philip’s arrival; the delay on the road risked this if the device of the invitation to Valladolid was to be carried out. For his plans it was essential that an irrevocable step should be taken in the king’s absence—a step which should compromise Carranza and commit the Inquisition so fully that Philip could not revoke it without damaging the Holy Office in a way that to him was impossible. To allow Carranza to be at liberty while investigating the suspicion of his heresy, as Philip had ordered, would leave the door open to royal or papal intervention; to seize and imprison him would leave Philip no alternative but to urge forward his destruction, while his dilatory progress could be assumed to cover preparations for flight. Accordingly, on August 17th the Suprema issued a commission, under the papal brief of January 7th, to Rodrigo de Castro to act with other inquisitors in the case, while, as justice required Carranza’s arrest, Valdés commissioned de Castro, Diego Ramírez and Diego González, inquisitor of Valladolid, to seize the person of the archbishop and convey him to such prison as should be designated, at the same time sequestrating all his property, real and personal and all his papers and writings. Simultaneously Joan Cebrian, alguazil mayor of the Suprema, was ordered to coöperate with the inquisitors in the arrest and sequestration.[181] Cebrian started the same day for Torrelaguna, where he kept his bed through the day and worked at night. The inquisitors came together; a force of familiars and others was secretly collected and, by day-break on the 22nd the governor, the alcalde and the alguaziles of Torrelaguna were seized and held under guard, the house where Carranza lodged was surrounded, de Castro, Ramírez, Cebrian and a dozen men ascended the stairs and knocked
- 73. ARCHBISHOP CARRANZA at the door of the antechamber. Fray Antonio de Utrilla asked who was there and the dread response came “Open to the Holy Office!” It was the same at the door of Carranza’s chamber; de Castro knelt at the bed-side, where Carranza had drawn the curtains and raised himself on his elbow; he begged Carranza’s pardon with tears in his eyes and said his face would show his reluctance in performing his duty. Cebrian was called in and read the order of arrest. Carranza replied “These señores do not know that they are not my judges, as I am subject directly to the pope.” Then de Castro produced the papal brief from the bosom of his gown and read it. Some say that Carranza fell back on his pillow, others that he remained imperturbable. He ordered out all the rest and remained for a considerable time alone with de Castro and Ramírez.[182] He was at once secluded in the most rigid manner, all his people being excluded, except Fray Domingo Ximenes, who was required to assist in the sequestration and inventory. At table he was served by de Castro and Ramírez, who treated him with the utmost respect and endeavored to console him, for by this time his fortitude had given way and he was overwhelmed. His attendants were all dismissed and given money to find their way whither they chose and their grief we are told moved every one to compassion. Only the cook and steward and a muleteer were retained to serve the party. At nine in the evening proclamation was made throughout the town that until daylight no one was to leave his house or look out of a window. At midnight Cebrian assembled forty horsemen; de Castro and Ramírez brought Carranza down and stationed themselves on either side of his mule as the cavalcade rode forth in the darkness and then Salinas, the owner of the house, was allowed to come out to close his door. The heat was overpowering and when, by ten in the morning they reached Lozoya, they rested for a day and a night. On the 27th they arrived at Laguna del Duero, near Valladolid, where de Castro and Ramírez left the party and rode forward for instructions, returning the same day and, at two in the morning of the 28th, Carranza was brought to the city and lodged in the house of Pedro González de Leon, in the suburb of San Pedro beyond the walls, which had been taken by the Inquisition.[183] Carranza thus disappeared from human sight as completely as though swallowed by the earth. It is a forcible illustration of inquisitorial methods, but
- 74. conspicuous only by reason of the dignity of the victim, for it rested with the discretion of the officials whether thus to spirit away and conceal their prisoners or to cast them publicly into the secret prison. Morales tells us that it was years before the place of Carranza’s incarceration was known, although every one said that he had been seized by the Holy Office. Even to say this, however, was not unattended with danger, for, in the trial, in September, by the tribunal of Toledo, of Rodrigo Alvárez, one of the charges against him was that, about September 5th, he had remarked to a casual fellow-traveller, that he came from Valladolid and was quite certain that the archbishop was imprisoned.[184] There could be no doubt about it in Toledo, where the news of the arrest was received on the 24th. On the 26th the chapter assembled in sorrow to take what measures they could, in aid of their beloved prelate, but they were powerless save to delegate two of their number to reside in Valladolid and render such assistance as was possible. It amounted to little save a testimony of sympathy, for no communication was allowed, but they advised with his counsel and performed what service they were able. This faithful watch was kept up during the long and weary years of the trial and when it was adjourned to Rome they went thither and remained to the end. The chapter also, almost monthly, sent memorials to Philip praying for a speedy and favorable end of the case. The great Dominican Order also felt keenly the disgrace inflicted on its distinguished member and exerted itself in his favor as far as it could. The Spanish episcopate also was greatly perturbed, not knowing where the next blow might fall and the scandal throughout the land was general.[185] Philip had disembarked at Laredo on August 29th. Valdés evidently felt that some excuse was necessary for action so much more decisive than that prescribed by the king and, in a letter of September 9th, explained to him that Carranza was delaying his movements in order to meet him on his arrival at Laredo; that he was working in Rome to impede the matter; that the infamy of his position was daily spreading and that the auto de fe prepared for the Lutherans could not take place while he was at liberty. Seeing that the effort to entice him to Valladolid had failed, it was resolved to bring him there, which was done quietly and without disturbance. He had been well treated and would continue to be so and the king might rely on the affair being conducted with all rectitude. An intimation, moreover, that
- 75. ARCHBISHOP CARRANZA all his property had been sequestrated indicates that the financial aspect of the matter was deemed worthy of being called to the royal attention and the whole tone of the letter shows that Carranza’s imprisonment was predetermined. The allusion to his design of meeting the king at Laredo disposes of the plea that he was suspected of flight and the fact that the auto de fe of the Lutherans did not take place until October 8th is a test of the flimsiness of the reasons alleged.[186] Carranza’s treatment was vastly better than that of ordinary prisoners confined in the cells of the secret prison. He was asked to select his attendants, when he named six, but was allowed only two—his companion, Fray Alonso de Utrilla and his page, Jorje Gómez Muñoz de Carrascosa. [187] Two rooms were allotted to the party—rooms without provision for the needs of human nature, with windows padlocked and shutters closed, so that at times the stench became unendurable. The foul atmosphere brought on a dangerous illness in which Carranza nearly perished; the physicians ordered the apartment to be ventilated, morning and evening, but all that the Suprema would permit was a small grating in the door, though at times it was left ajar with a guard posted at it.[188] Communication with the outside world was so completely cut off that when, in 1561, a great conflagration ravaged Valladolid, raging for thirty hours, destroying four hundred houses and penetrating to the quarter where the prison stood, the prisoners knew nothing of it until after reaching Rome.[189] The inquisitorial rule that all consultation with counsel must be held in the presence of an inquisitor was rigidly observed and also that which denied to prisoners the consolation of the sacraments. Diego González, one of the inquisitors of Valladolid, was assigned to the special charge of Carranza who, in a long and rambling memorial to the Suprema represents him as treating him without respect, insulting him, suppressing his communications with the Suprema, fabricating answers, throwing every impediment in the way of his defence and aggravating, with malicious ingenuity, the miseries of his position. Some details as to the parsimony with which he was treated are almost incredible when we reflect that the Inquisition and Philip were enjoying the enormous sequestrated revenues of their prisoner.[190]
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