<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%~MoulinÃ© par MaN_auto v.0.33.5 2025-03-11 15:50:49
\documentclass[CRMATH,Unicode,XML,biblatex]{cedram}

\TopicFR{Analyse fonctionnelle}
\TopicEN{Functional analysis}

\addbibresource{CRMATH_Houdayer_20240909.bib}

\usepackage{amssymb}
\usepackage{mathrsfs}
%\usepackage{mathtools}

%\newtheorem{letterthm}{Theorem}
%\renewcommand{\theletterthm}{\Alph{letterthm}}
%\newtheorem{lettercor}[letterthm]{Corollary}
%\renewcommand{\thelettercor}{\Alph{lettercor}}
%\newtheorem{letterapp}{Application}

\newtheorem*{CRC}{Connes' rigidity conjecture}

\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}

\DeclareMathOperator{\mult}{mult}
\DeclareMathOperator{\Prob}{Prob}
\DeclareMathOperator{\rk}{rk}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\rB}{B}
\DeclareMathOperator{\rE}{E}
\DeclareMathOperator{\rC}{C}
\DeclareMathOperator{\rL}{L}
\DeclareMathOperator{\Har}{Har}

\newcommand{\ovt}{\mathbin{\overline{\otimes}}}

\newcommand{\id}{\mathrm{id}}

\newcommand*{\dd}{\mathrm{d}}

%%%---------------------------------------------------------------------------

%%% ''Evaluated at'' macro
\newcommand\restr[2]{{% we make the whole thing an ordinary symbol
 \left.\kern-\nulldelimiterspace % automatically resize the bar with \right
 #1 % the function
 \vphantom{\big|} % pretend it's a little taller at normal size
 \right|_{#2} % this is the delimiter
 }}
% As a subscript: \phi_{F_f\rvert_{K_\ell}} [NO : \phi_{\left.h\right|_{K_\ell}}].

%%%---------------------------------------------------------------------------

%%% ARROWS

%%% \to macro
\renewcommand*{\to}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}
% Remplacer \xrightarrow{#1} par \overset{#1}{\longrightarrow}. 

%%% \mapsto macro
\let\oldmapsto\mapsto
\renewcommand*{\mapsto}{\mathchoice{\longmapsto}{\oldmapsto}{\oldmapsto}{\oldmapsto}}

%%% \longhookrightarrow macro
\newcommand{\longhookrightarrow}{\!\lhook\joinrel\longrightarrow}

%%%---------------------------------------------------------------------------

\hyphenation{iso-mor-phism}

\graphicspath{{./figures/}}

\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}

\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}

\newcommand*{\relabel}{\renewcommand{\labelenumi}{(\theenumi)}}
\newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}\relabel}
\newcommand*{\Romanenumi}{\renewcommand*{\theenumi}{\Roman{enumi}}\relabel}
\newcommand*{\alphenumi}{\renewcommand*{\theenumi}{\alph{enumi}}\relabel}
\newcommand*{\Alphenumi}{\renewcommand*{\theenumi}{\Alph{enumi}}\relabel}
\let\oldtilde\tilde
\renewcommand*{\tilde}[1]{\mathchoice{\widetilde{#1}}{\widetilde{#1}}{\oldtilde{#1}}{\oldtilde{#1}}}
\let\oldhat\hat
\renewcommand*{\hat}[1]{\mathchoice{\widehat{#1}}{\widehat{#1}}{\oldhat{#1}}{\oldhat{#1}}}

\title{Operator algebraic characterization of the noncommutative Poisson boundary}
\alttitle{CaractÃ©risation en algÃ¨bres d'opÃ©rateurs de la frontiÃ¨re de Poisson noncommutative}

\author{\firstname{Cyril} \lastname{Houdayer}}
\address{\'Ecole Normale Sup\'erieure, D\'epartement de Math\'ematiques et Applications, Universit\'e Paris-Saclay, 45 rue d'Ulm, 75230 Paris Cedex 05, France}
\email{cyril.houdayer@ens.psl.eu}
\thanks{The author is supported by ERC Advanced Grant NET 101141693.}
\CDRGrant[ERC Advanced Grant NET]{101141693}

\subjclass[2020]{20G25, 22D25, 46L10, 46L55}
\keywords{\kwd{Connes' rigidity conjecture} \kwd{higher rank lattices} \kwd{noncommutative Furstenberg--Poisson boundaries} \kwd{von Neumann algebras}}
\altkeywords{\kwd{Conjecture de rigiditÃ© de Connes} \kwd{rÃ©seaux de rang supÃ©rieur} \kwd{frontiÃ¨res de Furstenberg--Poisson noncommutatives} \kwd{algÃ¨bres de von Neumann}}

\begin{abstract}
We obtain an operator algebraic characterization of the noncommutative Furstenberg--Poisson boundary $\mathrm{L}(\Gamma) \subset \mathrm{L}(\Gamma \curvearrowright B)$ associated with an admissible probability measure $\mu \in \mathrm{Prob}(\Gamma)$ for which the $(\Gamma, \mu)$-Furstenberg--Poisson boundary $(B, \nu_B)$ is uniquely $\mu$-stationary. This is a noncommutative generalization of Nevo--Sageev's structure theorem~\cite{NS11}. We apply this result in combination with previous works to provide further evidence towards Connes' rigidity conjecture for higher rank lattices.
\end{abstract}

\begin{altabstract}
Nous obtenons une caractÃ©risation en algÃ¨bres opÃ©rateurs de la frontiÃ¨re de Furstenberg--Poisson noncommutative $\mathrm{L}(\Gamma) \subset \mathrm{L}(\Gamma \curvearrowright B)$ associÃ©e Ã&nbsp; une mesure de probabilitÃ© admissible $\mu \in \mathrm{Prob}(\Gamma)$ pour laquelle la $(\Gamma, \mu)$-frontiÃ¨re de Furstenberg--Poisson $(B, \nu_B)$ est uniquement $\mu$-stationnaire. Il s'agit d'une gÃ©nÃ©ralisation noncommutative du thÃ©orÃ¨me de structure de Nevo--Sageev~\cite{NS11}. Nous appliquons ce rÃ©sultat en combinaison avec des travaux antÃ©rieurs pour fournir des pistes supplÃ©mentaires afin de rÃ©soudre la conjecture de rigiditÃ© de Connes pour les rÃ©seaux de rang supÃ©rieur.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction and statement of the main results}

Let $\Gamma$ be a countable discrete group and $\mu \in \Prob(\Gamma)$ an admissible probability measure in the sense that $\bigcup_{n \geq 1} \supp(\mu)^n = \Gamma$. Denote by $(B, \nu_B)$ the $(\Gamma, \mu)$-Furstenberg--Poisson boundary~\cite{Fu62, BS04}. Recall that $(B, \nu_B)$ is the unique $(\Gamma, \mu)$-space, up to isomorphism, for which the $\Gamma$-equivariant Poisson transform
\[
\mathscr P_\mu \colon \rL^\infty(B, \nu_B) \to \Har^\infty(\Gamma, \mu) : f \mapsto \Biggl(\gamma \oldmapsto \int_B f(\gamma b) \, \dd \nu_B(b) \Biggr)
\]
is onto and isometric. Here, $\Har^\infty(\Gamma, \mu) \subset \ell^\infty(\Gamma)$ denotes the subspace of bounded (right) $\mu$-harmonic functions. We say that the $(\Gamma, \mu)$-Furstenberg--Poisson boundary $(B, \nu_B)$ is $\mu$-\emph{uniquely stationary} if there exists a compact metrizable model $K$ of $(B, \nu_B)$ for which there exists a unique $\mu$-stationary Borel probability measure $\nu_{K} \in \Prob(K)$ and $(K, \nu_{K}) \cong (B, \nu_B)$ as $(\Gamma, \mu)$-spaces. In~\cite[Theorem~9.2]{NS11}, Nevo--Sageev showed that if the $(\Gamma, \mu)$-Furstenberg--Poisson boundary $(B, \nu_B)$ is $\mu$-uniquely stationary, then for any amenable $(\Gamma, \mu)$-space $(X, \nu_X)$, there exists a relatively measure preserving $\Gamma$-equivariant measurable factor map $(X, \nu_X) \to (B, \nu_B)$.

Denote by $M = \rL(\Gamma)$ the group von Neumann algebra and by $\mathscr B = \rL(\Gamma \curvearrowright B)$ the group measure space construction associated with the nonsingular action $\Gamma \curvearrowright (B, \nu_B)$. Then we have the natural inclusion $M \subset \mathscr B$. We also consider the conjugation action $\Gamma \curvearrowright \mathscr B$ and we denote by $\rE_{B} \colon \mathscr B \to \rL^\infty(B, \nu_B)$ the canonical $\Gamma$-equivariant faithful normal conditional expectation. Then the faithful normal state $\psi_B = \nu_B \circ \rE_B \in \mathscr B_\ast$ is $\mu$-stationary. Regard $\rL^2(M, \tau) \subset \rL^2(\mathscr B, \psi_B)$ as a closed subspace and simply denote by $e_M \colon \rL^2(\mathscr B, \psi_B) \to \rL^2(M, \tau)$ the orthogonal projection. Define the normal state $\varphi_\mu \in \rB\bigl(\ell^2(\Gamma)\bigr)_\ast$ by the formula $\varphi_\mu(T) = \sum_{\gamma \in \Gamma} \mu(\gamma) \langle T \delta_\gamma, \delta_\gamma \rangle$ for $T \in \rB\bigl(\ell^2(\Gamma)\bigr)$. Define the normal ucp (unital completely positive) map
\[
\Phi_\mu \colon \rB\bigl(\ell^2(\Gamma)\bigr) \to \rB\bigl(\ell^2(\Gamma)\bigr) : T \mapsto \sum_{\gamma \in \Gamma} \mu(\gamma) J\lambda(\gamma)J T J \lambda(\gamma)^* J.
\]
Then we have $\varphi_\mu(T) = \bigl\langle \Phi_\mu(T)\delta_e, \delta_e \bigr\rangle$ for all $T \in \rB\bigl(\ell^2(\Gamma)\bigr)$. Denote by $\Har(\Phi_\mu) = \bigl\{T \in \rB\bigl(\ell^2(\Gamma)\bigr) \,\big\vert\, \Phi_\mu(T) = T\bigr\}$ the subspace of $\Phi_\mu$-harmonic elements. Following~\cite{Iz04, DP20}, the noncommutative Poisson transform
\[
\widehat{\mathscr P}_\mu \colon \mathscr B \to \Har(\Phi_\mu) : T \mapsto e_M T e_M
\]
is a normal onto isometric ucp map such that $\widehat{\mathscr P}_\mu\bigr\rvert_M = \id_M$. Then we refer to the inclusion $M \subset \mathscr B$ as the $(M, \varphi_\mu)$-noncommutative Furstenberg--Poisson boundary.

Let $M \subset \mathscr M$ be an inclusion of von Neumann algebras and consider the conjugation action $\Gamma \curvearrowright \mathscr M$. By~\cite[Proposition~4.2]{BBHP20}, to any (faithful) normal $\mu$-stationary state $\psi \in \mathscr M_\ast$ corresponds a unique $\Gamma$-equivariant (faithful) normal ucp map $\Theta \colon \mathscr M \to \rL^\infty(B, \nu_B)$ such that $\nu_B \circ \Theta = \psi$. Moreover, by~\cite[Proposition~2.1]{DP20}, to any (faithful) normal $\mu$-stationary state $\psi \in \mathscr M_\ast$ corresponds a unique (faithful) normal ucp map $\widehat \Theta \colon \mathscr M \to \mathscr B$ such that $\restr{\widehat \Theta}{M} = \id_M$, $\psi = \nu_B \circ \rE_B \circ\: \widehat\Theta$ and $\Theta = \rE_B \circ \: \widehat \Theta$.

Whenever $\Phi \colon A \to B$ is a ucp map between unital $\rC^*$-algebras, we denote by $\mult(\Phi) \subset A$ the multiplicative domain of $\Phi$. Our main result is the following noncommutative generalization of Nevo--Sageev's structure theorem~\cite[Theorem~9.2]{NS11}.

\begin{theo}\label{thm-NCNS}
Keep the same notation as above with $M = \rL(\Gamma)$ and $\mathscr B = \rL(\Gamma \curvearrowright B)$. Assume that the $(\Gamma, \mu)$-Furstenberg--Poisson boundary $(B, \nu_B)$ is $\mu$-uniquely stationary. Let $M \subset \mathscr M$ be an inclusion of von Neumann algebras. Assume that $\mathscr M$ is amenable and that there exists a faithful normal ucp map $\widehat\Theta \colon \mathscr M \to \mathscr B$ such that $\restr{\widehat \Theta}{M} = \id_M$.

Then $\restr{\widehat\Theta}{\mult(\widehat\Theta)} \colon \mult(\widehat\Theta) \to \mathscr B$ is a unital normal onto $\ast$-isomorphism. Therefore, we may regard $\mathscr B \subset \mathscr M$ as a von Neumann subalgebra and $\widehat \Theta\colon \mathscr M \to \mathscr B$ as a faithful normal conditional expectation.
\end{theo}

In particular, letting $\psi = \nu_B \circ \rE_B \circ\: \widehat \Theta \in \mathscr M_\ast$ and $\psi_B = \nu_B \circ \rE_B \in \mathscr B_\ast$, it follows from~\cite[Theorem~A]{Zh23} and~\cite{KV82} that the inclusion $M \subset \mathscr M$ has maximal $\varphi_\mu$-entropy in the sense of~\cite{DP20}, that is,
\[
h_{\varphi_\mu}\bigl(M \subset \mathscr M, \psi\bigr) = h_{\varphi_\mu}\bigl(M \subset \mathscr B, \psi_B\bigr) = h(\mu) = \lim_n \frac{1}{n} H(\mu^{\ast n}).
\]

When $\Gamma$ is moreover an infinite icc group (infinite conjugacy classes), we deduce the following operator algebraic characterization of the $(M, \varphi_\mu)$-noncommutative Furstenberg--Poisson boundary $M \subset \mathscr B$.

\begin{coro}\label{cor-NCNS}
Keep the same notation as above with $M = \rL(\Gamma)$ and $\mathscr B = \rL(\Gamma \curvearrowright B)$. Assume that $\Gamma$ is infinite icc and that the $(\Gamma, \mu)$-Furstenberg--Poisson boundary $(B, \nu_B)$ is $\mu$-uniquely stationary. Let $M \subset \mathscr M$ be an inclusion of von Neumann algebras satisfying the following conditions:
\begin{enumerate}\romanenumi
\item \label{cor-NCNS_1} $M' \cap \mathscr M = \C1$.
\item \label{cor-NCNS_2} There exists a normal ucp map $\widehat \Theta \colon \mathscr M \to \mathscr B$ such that $\restr{\widehat\Theta}{M} = \id_{M}$.
\item \label{cor-NCNS_3} $\mathscr M$ is amenable.
\item \label{cor-NCNS_4} Whenever $M \subset \mathscr N \subset \mathscr M$ is an intermediate von Neumann subalgebra for which there exists a normal conditional expectation $\rE \colon \mathscr M \to \mathscr N$, we have $\mathscr N = \mathscr M$.
\end{enumerate}
Then $\widehat \Theta \colon \mathscr M \to \mathscr B$ is a unital normal onto $\ast$-isomorphism.
\end{coro}

Let us point out that the $(M, \varphi_\mu)$-noncommutative Furstenberg--Poisson boundary $M \subset \mathscr B$ satisfies Conditions~\eqref{cor-NCNS_1}, \eqref{cor-NCNS_2}, \eqref{cor-NCNS_3}, \eqref{cor-NCNS_4} by~\cite{DP20}. Thus, Corollary~\ref{cor-NCNS} gives an abstract operator algebraic characterization of the inclusion $M \subset \mathscr B$.

We also apply Theorem~\ref{thm-NCNS} to provide further evidence towards Connes' rigidity conjecture for higher rank lattices.

\begin{CRC}
For every $i \in \{1, 2\}$, let $G_i$ be a semisimple connected real Lie group with trivial center, no compact factor such that\/ $\rk_{\R}(G_i) \geq 2$ and let $\Gamma_i &lt; G_i$ be an irreducible lattice. If\/ $\rL(\Gamma_1) \cong \rL(\Gamma_2)$, then $G_1 \cong G_2$ and in particular $\rk_{\R}(G_1) = \rk_{\R}(G_2)$.
\end{CRC}

For every $i \in \{1, 2\}$, choose a Furstenberg measure $\mu_i \in \Prob(\Gamma_i)$ so that $(G_i/P_i, \nu_{P_i})$ is the $(\Gamma_i, \mu_i)$-Furstenberg--Poisson boundary~\cite{Fu67}. Here, $\nu_{P_i} \in \Prob(G_i/P_i)$ denotes the unique $K_i$-invariant Borel probability measure, where $K_i &lt; G_i$ is a maximal compact subgroup and $G_i = K_i P_i$. It is well-known that $(G_i/P_i, \nu_{P_i})$ is $\mu_i$-uniquely stationary~\cite{GM89}. Set $M_i = \rL(\Gamma_i)$ and $\mathscr B_i = \rL(\Gamma_i \curvearrowright G_i/P_i)$.

Assume that $M_1 \cong M_2$ and set $M = M_1 = M_2$. Since $\mathscr B_2$ is an amenable (hence injective) von Neumann algebra and since $M \subset \mathscr B_1$, Arveson's extension theorem implies that there exists a ucp map $\Phi \colon \mathscr B_1 \to \mathscr B_2$ such that $\restr{\Phi}{M} = \id_M$. We show that if $\Phi \colon \mathscr B_1 \to \mathscr B_2$ is normal, then $\rk_\R(G_1) = \rk_\R(G_2)$.

\begin{theo}\label{thm-CRC}
Keep the same notation as above. Assume that there exists a normal ucp map $\Phi \colon \mathscr B_1 \to \mathscr B_2$ such that $\restr{\Phi}{M} = \id_M$. Then $\Phi \colon \mathscr B_1 \to \mathscr B_2$ is a unital normal onto $\ast$-isomorphism. In particular, we have
\[
\rk_\R(G_1) = \rk_\R(G_2).
\]
\end{theo}

\section{Proofs of the main results}

We record the following well-known fact on ucp maps.

\begin{lemm}\label{lem-useful1}
Let\/ $\mathfrak A, \mathfrak B, \mathfrak C$ be unital $\rC^*$-algebras and $\Phi \colon \mathfrak A \to \mathfrak B$, $\Psi \colon \mathfrak B \to \mathfrak C$ be ucp maps. Assume that $\Psi$ is faithful and that $\Psi \circ \Phi$ is a unital $\ast$-homomorphism. Then $\Phi$ is a unital $\ast$-homomorphism and $\Phi(\mathfrak A) \subset \mult(\Psi)$.
\end{lemm}

\begin{proof}
By Kadison's inequality and since $\Psi \circ \Phi$ is a unital $\ast$-homomorphism, for every $a \in \mathfrak A$, we have
\[
\Psi\bigl(\Phi(a^*a)\bigr) = \Psi\bigl(\Phi(a^*)\bigr)\Psi\bigl(\Phi(a)\bigr) \leq \Psi\bigl(\Phi(a)^* \Phi(a)\bigr) \leq \Psi\bigl(\Phi(a^*a)\bigr).
\]
Since $\Psi$ is faithful, it follows that $\Phi(a)^* \Phi(a) = \Phi(a^*a)$. Then $\mult(\Phi) = \mathfrak A$ and so $\Phi$ is a unital $\ast$-homomorphism. Moreover, for every $a \in \mathfrak A$, we have
\[
\Psi\bigl(\Phi(a)^*\bigr)\Psi\bigl(\Phi(a)\bigr) = \Psi\bigl(\Phi(a)^* \Phi(a)\bigr).
\]
This further implies that $\Phi(\mathfrak A) \subset \mult(\Psi)$.
\end{proof}

Let $\Gamma$ be a countable discrete group and $\mu \in \Prob(\Gamma)$ an admissible probability measure. Denote by $(B, \nu_B)$ the $(\Gamma, \mu)$-Furstenberg--Poisson boundary. Set $M = \rL(\Gamma)$ and $\mathscr B = \rL({\Gamma \curvearrowright B})$. Denote by $\rE_{B} \colon \mathscr B \to \rL^\infty(B, \nu_B)$ the canonical $\Gamma$-equivariant faithful normal conditional expectation.

Let $M \subset \mathscr M$ be an inclusion of von Neumann algebras. Let $\widehat\Theta\colon \mathscr M \to \mathscr B$ be a faithful normal ucp map such that $\restr{\widehat \Theta}{M} = \id_M$. Then $\Theta = \rE_B \circ\: \widehat \Theta \colon \mathscr M \to \rL^\infty(B, \nu_B)$ is a $\Gamma$-equivariant faithful normal ucp map.

\begin{lemm}\label{lem-useful2}
Keep the same notation as above. We have
\[
\Theta\bigl(\mult(\Theta)\bigr) \rtimes \Gamma \subset \widehat\Theta\bigl(\mult(\widehat\Theta)\bigr) \subset \rL^\infty(B, \nu_B) \rtimes \Gamma =\mathscr B.
\]
In particular, if\/ $\Theta\bigl(\mult(\Theta)\bigr) = \rL^\infty(B, \nu_B)$, then $\widehat\Theta\bigl(\mult(\widehat\Theta)\bigr) = \mathscr B$. In that case, we may regard $\mathscr B \subset \mathscr M$ as a von Neumann subalgebra and $\widehat \Theta \colon \mathscr M \to \mathscr B$ as a faithful normal conditional expectation.
\end{lemm}

\begin{proof}
By definition, $\restr{\Theta}{\mult(\Theta)} \colon \mult(\Theta) \to \rL^\infty(B, \nu_B)$ is a $\Gamma$-equivariant unital normal $\ast$-isomorphism and $\restr{\widehat\Theta}{\mult(\widehat\Theta)} \colon \mult(\widehat\Theta) \to \mathscr B$ is a unital normal $\ast$-isomorphism such that $M \subset \widehat\Theta\bigl(\mult(\widehat\Theta)\bigr)\subset \mathscr B$. Since $\rE_B \circ\: \widehat \Theta = \Theta$ and since $\rE_B$ is faithful, Lemma~\ref{lem-useful1} implies that $\restr{\widehat{\Theta}}{\mult(\Theta)} \colon \mult(\Theta) \to \mathscr B$ is a unital normal $\ast$-isomorphism and $\widehat \Theta\bigl(\mult(\Theta)\bigr) \subset \mult(\rE_B)$. Then we have $\mult(\Theta) \subset \mult(\widehat\Theta)$ and so $M \vee \mult(\Theta) \subset \mult(\widehat \Theta)$. Since $\mult(\rE_B) = \rL^\infty(B, \nu_B) \subset \mathscr B$, it follows that $\widehat\Theta\bigl(\mult(\Theta)\bigr) \subset \rL^\infty(B, \nu_B) \subset \mathscr B$ and that $\restr{\Theta}{\mult(\Theta)} = \restr{\widehat \Theta}{\mult(\Theta)}$. Then we have
\[
\Theta\bigl(\mult(\Theta)\bigr) \rtimes \Gamma \subset \widehat\Theta\bigl(\mult(\widehat\Theta)\bigr) \subset \rL^\infty(B, \nu_B) \rtimes \Gamma = \mathscr B.
\]

Moreover, assume that $\Theta\bigl(\mult(\Theta)\bigr) = \rL^\infty(B, \nu_B)$. Then we necessarily have $\widehat\Theta\bigl(\mult(\widehat\Theta)\bigr) = \mathscr B$. Define the unital normal $\ast$-isomorphism $\iota = (\restr{\widehat\Theta}{\mult(\widehat \Theta)})^{-1} \colon \mathscr B \to \mult(\widehat\Theta)$. Then $\iota \circ \Theta \colon \mathscr M \to \mult(\widehat \Theta)$ is a faithful normal conditional expectation.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm-NCNS}]
Keep the same notation as above with $M = \rL(\Gamma)$ and $\mathscr B = \rL(\Gamma \curvearrowright B)$. Assume moreover that the $(\Gamma, \mu)$-Furstenberg--Poisson boundary $(B, \nu_B)$ is $\mu$-uniquely stationary. We still denote by $B$ the compact metrizable model and we assume that $B = \supp(\nu_B)$. Then the identity map $\id_{\rC(B)} \colon \rC(B) \to \rL^\infty(B, \nu_B)$ is the unique $\Gamma$-equivariant ucp map (see~\cite[Corollary~VI.2.10]{Ma91} and~\cite[Theorem~3.4]{HK21}). Denote by $\lambda \colon \Gamma \to \mathscr U(M) : \gamma \mapsto \lambda(\gamma)$ the left regular representation. We naturally identify $\rL^2(M) = \ell^2(\Gamma)$. Denote by $M'$ the commutant of $M$ in $\rB\bigl(\rL^2(M)\bigr)$. Since $M \subset \mathscr M$, we may regard $\rL^2(\mathscr M)$ as a Hilbert left $M$-module. Then~\cite[Proposition~8.2.3]{AP14} implies that there exists a projection $e \in M' \ovt \rB(\ell^2)$ and a unitary mapping $V \colon \rL^2(\mathscr M) \to e\bigl(\rL^2(M) \otimes \ell^2\bigr)$ such that $V(a \xi) = (a \otimes 1) V(\xi)$ for every $a \in M$ and every $\xi \in \rL^2(\mathscr M)$. Then the mapping
\[
\pi \colon \rB\bigl(\rL^2(M)\bigr) \to \rB\bigl(\rL^2(\mathscr M)\bigr) : T \mapsto V^* e(T \otimes 1) eV
\]
is a normal ucp map such that $\pi(a T b) = a \pi(T) b$ for all $T \in \rB\bigl(\rL^2(M)\bigr)$ and all $a, b \in M$.

Since $\mathscr M \subset \rB\bigl(\rL^2(\mathscr M)\bigr)$ is amenable, there exists a (possibly non normal) conditional expectation $\rE \colon \rB\bigl(\rL^2(\mathscr M)\bigr) \to \mathscr M$. Choose a point $b \in B$ and define the $\Gamma$-equivariant unital $\ast$-homomorphism $\theta \colon \rC(B) \to \ell^\infty(\Gamma) : F \mapsto \bigl(\gamma \mapsto F(\gamma b)\bigr)$. Regard $\ell^\infty(\Gamma) \subset \rB\bigl(\ell^2(\Gamma)\bigr) = \rB\bigl(\rL^2(M)\bigr)$ and define the ucp map $\iota = \rE \circ\, \pi \circ \theta \colon \rC(B) \to \mathscr M$. Then for every $F \in \rC(B)$ and every $\gamma \in \Gamma$, we have
\[
\iota(F \circ \gamma^{-1}) = \rE\Bigl(\pi\bigl(\theta(F) \circ \gamma^{-1}\bigr)\Bigr) = \rE\Bigl(\pi\bigl(\lambda(\gamma) \theta(F) \lambda(\gamma)^\ast\bigr)\Bigr) = \lambda(\gamma) \iota(F) \lambda(\gamma)^*.
\]
Therefore, the ucp map $\iota \colon \rC(B) \to \mathscr M$ is $\Gamma$-equivariant with respect to the conjugation action $\Gamma \curvearrowright \mathscr M$.

By composition, $\Theta \circ \iota \colon \rC(B) \to \rL^\infty(B, \nu_B)$ is a $\Gamma$-equivariant ucp map. Then we have $\Theta \circ \iota = \id_{\rC(B)}$. Since $\Theta \colon \mathscr M \to \rL^\infty(B, \nu_B)$ is faithful, Lemma~\ref{lem-useful1} implies that $\iota \colon \rC(B) \to \mathscr M$ is a unital $\ast$-homomorphism and $\iota\bigl(\rC(B)\bigr) \subset \mult(\Theta)$. Set $\psi = \nu_B \circ \Theta \in \mathscr M_\ast$ and note that $\psi$ is a $\mu$-stationary faithful normal state. Since $\psi \circ \iota = \nu_B$, we may uniquely extend $\iota \colon \rL^\infty(B, \nu_B) \to \mathscr M$ to a faithful normal unital $\ast$-homomorphism such that $\Theta \circ \iota = \id_{\rL^\infty(B, \nu_B)}$. Then we have $\iota\bigl(\rL^\infty(B, \nu_B)\bigr) \subset \mult(\Theta)$, $\rL^\infty(B, \nu_B) = \Theta\bigl(\iota\bigl(\rL^\infty(B, \nu_B)\bigr)\bigr)$ and so $\Theta\bigl(\mult(\Theta)\bigr) = \rL^\infty(B, \nu_B)$. The moreover part follows from Lemma~\ref{lem-useful2}.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor-NCNS}]
Set $M = \rL(\Gamma)$ and $\mathscr B = \rL(\Gamma \curvearrowright B)$. Assume that $\Gamma$ is infinite icc and that the $(\Gamma, \mu)$-Furstenberg--Poisson boundary $(B, \nu_B)$ is $\mu$-uniquely stationary. Denote by $\lambda \colon \Gamma \to \mathscr U(M) : \gamma \mapsto \lambda(\gamma)$ the left regular representation. Let $\widehat \Theta \colon \mathscr M \to \mathscr B$ be a normal ucp map such that $\restr{\widehat\Theta}{M} = \id_{M}$ as given by Condition~\eqref{cor-NCNS_3}. Denote by $p \in \mathscr M$ the support projection of $\widehat \Theta \colon \mathscr M \to \mathscr B$. Then for $\gamma \in \Gamma$, we have $\widehat\Theta\bigl(\lambda(\gamma) p \lambda(\gamma)^*\bigr) = \lambda(\gamma) \widehat\Theta(p) \lambda(\gamma)^* = 1$ and so $p \leq \lambda(\gamma) p \lambda(\gamma)^*$. Since this holds for every $\gamma \in \Gamma$, we infer that $p \in \rL(\Gamma)'\cap \mathscr M$. Condition~\eqref{cor-NCNS_1} further implies that $p = 1$. Thus, $\widehat \Theta \colon \mathscr M \to \mathscr B$ is a faithful normal ucp map. By Condition~\eqref{cor-NCNS_2} and Theorem~\ref{thm-NCNS}, we infer that $\mathscr B \subset \mathscr M$ can be regarded as a von Neumann subalgebra and $\widehat\Theta \colon \mathscr M \to \mathscr B$ as a faithful normal conditional expectation. Then Condition~\eqref{cor-NCNS_4} finally implies that $\mathscr B = \mathscr M$ and so $\widehat\Theta \colon \mathscr M \to \mathscr B$ is a unital normal onto $\ast$-isomorphism.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm-CRC}]
Consider the inclusion $M \subset \mathscr B_1$. Since $M$ is a type $\mathrm{II}_1$ factor, \cite[Proposition~2.7]{DP20} implies that $M' \cap \mathscr B_1 = \C 1$. By assumption, $\mathscr B_1$ is amenable and $\Phi \colon \mathscr B_1 \to \mathscr B_2$ is a normal ucp map such that $\restr{\Phi}{M} = \id_M$. By~\cite[Theorem~4.1]{DP20}, for every intermediate von Neumann subalgebra $M \subset \mathscr N \subset \mathscr B_1$ for which there exists a normal conditional expectation $\rE \colon \mathscr B_1 \to \mathscr N$, we have $\mathscr N = \mathscr B_1$. We may apply Corollary~\ref{cor-NCNS} to the inclusion $M \subset \mathscr B_1$ to conclude that $\Phi \colon \mathscr B_1 \to \mathscr B_2$ is a unital normal onto $\ast$-isomorphism. By~\cite[Corollary~F]{Ho21} and~\cite[Theorem~B]{BH22}, we infer that $\rk_{\R}(G_1) = \rk_{\R}(G_2)$.
\end{proof}

\section*{Acknowledgments}
I am grateful to Amos Nevo for thought provoking discussions during the IHP trimester \emph{Group actions and rigidity: around the Zimmer program} and for pointing out the reference~\cite{NS11}.

\section*{Declaration of interests}
The author does not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and has declared no affiliations other than their research organizations.

\printbibliography
\end{document}</pre></body></html>