On the weak relative Dixmier property

TL;DR

This paper proves that all inclusions of von Neumann algebras with a faithful normal conditional expectation possess the weak relative Dixmier property, resolving a question posed by Popa.

Contribution

It establishes the weak relative Dixmier property for all such inclusions, using an improved version of Ellis' lemma for compact convex semigroups.

Findings

All inclusions of von Neumann algebras with a faithful normal conditional expectation have the weak relative Dixmier property.

Provides an affirmative answer to Popa's question from 1999.

Introduces an improved version of Ellis' lemma for compact convex semigroups.

Abstract

We show that every inclusion of von Neumann algebras with a faithful normal conditional expectation has the weak relative Dixmier property. This answers a question of Popa \cite{Po99}. The proof uses an improvement of Ellis' lemma for compact convex semigroup.

Equations

$$E\circ \Phi =\Phi \circ E={\Phi }_0\circ \Psi \circ E={\Phi }_0\circ \phi =\phi .$$

Full text

The author is supported by JSPS

1. Introduction

Let $N\subset M$ be an inclusion of von Neumann algebras. Following [Po99, Definition 1.1], we say that the inclusion $N\subset M$ has the weak relative Dixmier property if for every $x\in M$, the weak∗ closed convex hull of $\{uxu^{*}\mid u\in\mathcal{U}(N)\}$ intersects $N^{\prime}\cap M$. A classical result of Dixmier asserts that this property is always satisfied when $N=M$. However, as explained in [HP17], if one does not make any assumption on the inclusion $N\subset M$, then the weak relative Dixmier property is not necessarily satisfied and can be very subtle as demonstrated by the following two examples:

• •

The inclusion $M\subset\mathbf{B}(H)$ has the weak relative Dixmier property if and only if $M$ is AFD. The if direction was first noticed by Schwartz [Sc63] while the converse follows from Connes’ celebrated work [Co75].

• •

A type ${\rm III}_{1}$ factor $M$ has a trivial bicentralizer if and only if the inclusion $N\subset M$ has the weak relative Dixmier property, where $N$ is the continuous core of $M$. This equivalence is due to Haagerup [Ha85]. It is still a deep open problem to determine whether every type ${\rm III}_{1}$ factor has a trivial bicentralizer.

On the other hand, it is an elementary fact [Po99, Lemma 1.2] that an inclusion $N\subset M$ has the weak relative Dixmier property if $M$ is tracial or more generally if there exists a faithful normal state $\varphi$ on $M$ such that $N\subset M_{\varphi}:=\{a\in M\mid\forall x\in M,\>\varphi(ax)=\varphi(xa)\}$. This kind of averaging argument is actually a fundamental tool in von Neumann algebra theory. For example, it plays a key role in [Po81] or in Popa’s intertwining theory [Po01, Po03]. In [Po99, Remark 1.5] (see also [HP17, Problem 7]), it was suggested that the weak relative Dixmier property might more generally hold for any inclusion with expectation (this would in particular cover [Po99, Corollary 1.4]). Our main theorem solves this question.

Main theorem**.**

Let $N\subset M$ be an inclusion of von Neumann algebras. Suppose that there exists a faithful normal state $\varphi$ on $M$ such that $N$ is the range of a $\varphi$-preserving conditional expectation $E:M\rightarrow N$. Then the inclusion $N\subset M$ has the weak relative Dixmier property.

The difficulty in the proof of this theorem arises only when $N$ is of type ${\rm III}_{1}$. Indeed, in all other cases, one can use discrete decompositions in order to reduce the problem to the tracial case. To deal with the remaining type ${\rm III}_{1}$ case, we use a recent result of the author [Ma18] asserting the existence of an irreducible AFD subfactor with expectation inside $N$. We combine it with a new averaging argument for compact convex semigroups based on Ellis’ lemma and inspired by Sinclair’s proof of the existence of the injective enveloppe [Si15].

2. Compact convex semigroups

Definition 2.1**.**

A compact semigroup $K$ is a non-empty compact space equipped with a semigroup operation such that $K\ni a\mapsto ab\in K$ is continuous for every given $b\in K$.

The proof of the following result, known as Ellis’ lemma, is very short but it has many surprising applications to combinatorics [FK89]. Recently, Sinclair applied it to give a very short proof of the existence of the injective enveloppe of an operator system [Si15].

Recall that an element $e$ of a semigroup $S$ is idempotent if $ee=e$. We define an order relation between idempotents as follows: $e<f$ if and only if $ef=fe=e$.

Theorem 2.2** ([FK89]).**

Let $K$ be a compact semigroup. Then $K$ contains an idempotent which is minimal for the order relation $<$.

Remark 2.3**.**

It is also shown in [FK89] that an idempotent $e\in K$ is minimal if and only if $Ke$ is a minimal closed left ideal.

Definition 2.4**.**

A compact convex semigroup $K$ is a non-empty compact convex space (in some locally convex topological vector space) equipped with a semigroup operation such that $K\ni a\mapsto ab\in K$ is affine and continuous for every given $b\in K$.

Example 2.5**.**

Let $M$ be a von Neumann algebra and let $\mathrm{UCP}(M)$ be the set of all UCP maps from $M$ into $M$, equipped with the topology of pointwise weak∗ convergence. Then $\mathrm{UCP}(M)$ is a compact convex semigroup.

We now strengthen the conclusion of Theorem 2.2 for compact convex semigroups.

Theorem 2.6**.**

Let $K$ be a compact convex semigroup. Let $e\in K$ be a minimal idempotent. Then we have $exe=e$ for all $x\in K$.

Proof.

Let $I=Ke$ be the minimal closed left ideal generated by $e$. Take any $x\in K$ and let $J=\{\frac{1}{2}(bxe+b)\mid b\in I\}$. Then $J\subset I$ and $J$ is a closed left ideal of $K$. By minimality of $I$, we must have $J=I$. Let $a$ be an extremal point of the convex compact space $I$. Since $J=I$, we can find $b\in I$, such that $a=\frac{1}{2}(bxe+b)$. Because $a$ is extremal, this forces $a=bxe=b$. Thus $axe=a$. Since this holds for every extremal point $a\in I$, and since $I$ is the closed convex hull of its extremal points (Krein-Millmann theorem), we conclude that $axe=a$ for all $a\in I$. In particular, we have $exe=e$ as we wanted. ∎

Recall that if $A$ is a unital $C^{*}$-algebra and $\Phi:A\rightarrow A$ is an idempotent UCP map, then $\Phi(A)$ equipped with the Choi-Effros product $(x,y)\mapsto\Phi(xy)$ is a $C^{*}$-algebra [CE77, Theorem 3.1]. By combining this fact with Theorem 2.6, one can associate to each compact convex semigroup of UCP maps on a von Neumann algebra a canonical “boundary” $C^{*}$-algebra. See [Si15] for examples.

Corollary 2.7**.**

Let $M$ be a von Neumann and $K\subset\mathrm{UCP}(M)$ a closed convex semigroup. Let $\Phi$ and $\Psi$ be two minimal idempotents. Then $\Psi|_{\Phi(M)}:\Phi(M)\rightarrow\Psi(M)$ is an isomorphism of $C^{*}$-algebras with inverse $\Phi|_{\Psi(M)}$.

Proof.

By Theorem 2.6, we have $\Psi\circ\Phi\circ\Psi=\Psi$ and $\Phi\circ\Psi\circ\Phi=\Phi$. Terefore $\Psi|_{\Phi(M)}$ and $\Phi|_{\Psi(M)}$ are UCP maps which are inverse to each others. Thus they must be isomorphisms of $C^{*}$-algebras. ∎

Proposition 2.8**.**

Let $M$ be a von Neumann and $K\subset\mathrm{UCP}(M)$ a closed convex semigroup. Let $M^{K}=\{x\in M\mid\forall\Psi\in K,\>\Psi(x)=x\}$. Let $\Phi\in K$ be a minimal idempotent. If $\Phi$ is faithful, then $M^{K}$ is a $C^{*}$-subalgebra of $M$ and $\Phi$ is a conditional expectation of $M$ onto $M^{K}$.

Proof.

For every $x\in\Phi(M)$, we have $\Phi(x^{*}x)\geq\Phi(x^{*})\Phi(x)=x^{*}x$. But we have $\Phi(\Phi(x^{*}x)-x^{*}x)=0$. Thus $\Phi(x^{*}x)=x^{*}x$ because $\Phi$ is faithful. In particular, $x^{*}x\in\Phi(M)$ for all $x\in\Phi(M)$. By the polarization identity, this means that $\Phi(M)$ is a subalgebra of $M$. Then $\Phi$ is a faithful conditional expectation onto $\Phi(M)$. Now, take a unitary $u\in\Phi(M)$. For every $\Psi\in K$, we have $\Phi(\Psi(u))=u$ by Theorem 2.6. Since $\Phi$ is a faithful conditional expectation, this forces $\Psi(u)=u$. Thus $u\in M^{K}$. Since this holds for every unitary $u\in\Phi(M)$, we conclude that $\Phi(M)=M^{K}$. ∎

3. The weak relative Dixmier property

Let $N\subset M$ be an inclusion of von Neumann algebras. Let $\mathcal{D}(N,M)\subset\mathrm{UCP}(M)$ be the closed convex semigroup generated by $\operatorname{Ad}(u)$ for all $u\in\mathcal{U}(N)$. Note that $M^{\mathcal{D}(N,M)}=N^{\prime}\cap M$. Observe that the inclusion $N\subset M$ has the weak relative Dixmier property if and only if $\mathcal{D}(N,M)$ contains a conditional expectation onto $N^{\prime}\cap M$.

We need the following key lemma which relies on [Ma18, Theorem D].

Lemma 3.1**.**

Let $N$ be a type ${\rm III}_{1}$ factor with separable predual. Let $\varphi$ be a faithful normal state on $N$. Then $\varphi\in\mathcal{D}(N,N)$.

Proof.

By [Ma18, Theorem D], $N$ contains an irreducible subfactor with expectation $P\subset N$ such that $P$ is AFD of type ${\rm III}_{1}$. Let $\psi$ be a faithful normal state on $N$ such that $P$ is globally invariant by $\sigma^{\psi}$ and $P_{\psi}^{\prime}\cap P=\mathbf{C}$. Since $P$ is AFD and irreducible in $N$, we know that $\mathcal{D}(P,N)$ contains a state $\phi$. Note that $\phi=\phi\circ E^{\psi}_{P}$. Thus, we have $\phi\circ E^{\psi}_{P_{\psi}^{\prime}\cap N}=\psi$. Since $P_{\psi}$ centralizes $\psi$, we have $E^{\psi}_{P_{\psi}^{\prime}\cap N}\in\mathcal{D}(P_{\psi},N)$. We conclude that $\psi=\phi\circ E^{\psi}_{P_{\psi}^{\prime}\cap N}\in\mathcal{D}(P,N)\subset\mathcal{D}(N,N)$. Finally, by Connes-Størmer transitivity theorem, we get $\varphi\in\mathcal{D}(N,N)$. ∎

We are now ready to prove our main theorem.

Proof of the main theorem.

First note that we can always write $N$ as an increasing union of a net of subalgebras $(N_{i})_{i\in I}$ with separable predual which are globally invariant under $\sigma^{\varphi}$. Thus, we may assume that $N$ has a separable predual. Then, we may also assume that $M$ has a separable predual. Since $\mathcal{Z}(N)$ has the weak relative dixmier property in $M$, we may replace $M$ by $\mathcal{Z}(N)^{\prime}\cap M$ and reduce to the case where $\mathcal{Z}(N)\subset\mathcal{Z}(M)$. Now by using the desintegration theory, we may assume that $N$ is a factor. If $N$ is semifinite, the result is already known. So we may assume that $N$ is a type ${\rm III}$ factor. If $N$ is not of type ${\rm III}_{1}$, then $N\cong N_{0}\rtimes\mathbf{Z}$ for some semifinite von Neumann algebra with expectation $N_{0}\subset N$. Since $N_{0}$ has the weak relative dixmier property in $M$ and $\mathbf{Z}$ is amenable, then $N$ also has the weak relative dixmier property in $M$.

Finally, we deal with the case where $N$ is of type ${\rm III}_{1}$. By compactness of $\mathcal{D}(N,M)$, any element of $\mathcal{D}(N,N)$ can be extended to an element of $\mathcal{D}(N,M)$. In particular, thanks to Lemma 3.1, we can find $\Psi\in\mathcal{D}(N,M)$ such that $\Psi|_{N}=\varphi$. Take $\Phi_{0}$ a minimal idempotent in $\mathcal{D}(N,M)$ and let $\Phi=\Phi_{0}\circ\Psi$ which is again a minimal idempotent of $\mathcal{D}(N,M)$. Observe that $E$ commutes with every element of $\mathcal{D}(N,M)$. Thus, we have

[TABLE]

In particular, $\Phi$ is faithful and by Proposition 2.8, we conclude that $\Phi$ is a conditional expectation on $N^{\prime}\cap M$ (in fact $\Phi$ is the unique $\varphi$-preserving conditional expectation onto $N^{\prime}\cap M$). ∎

Remark 3.2**.**

Let $M\subset\mathbf{B}(H)$ be a von Neumann algebra. Let $\Phi\in\mathcal{D}(M,\mathbf{B}(H))$ be a minimal idempotent. Then $A_{\Phi}=\Phi(\mathbf{B}(H))$ (whose isomorphism class as a $C^{*}$-algebra does not depend on the choice of $\Phi$) contains $M^{\prime}$. We have $A_{\Phi}=M^{\prime}$ if and only if $M$ is injective. In general, $A_{\Phi}$ contains the injective enveloppe of $M^{\prime}$ but it is not clear if they are equal or not. Notice that if $M\subset N\subset\mathbf{B}(H)$, then $M\subset N$ has the weak relative Dixmier property if and only if $A_{\Phi}\cap N=M^{\prime}\cap N$.