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\datereceived{2022-07-28}
\dateaccepted{2023-02-20}
\dateepreuves{2023-02-24}

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\datepublished{2023-02-28}
\begin{document}
\frontmatter

\title[The noncommutative factor theorem for lattices in product groups]{The noncommutative factor theorem for\\ lattices in product groups}

\author[\initial{R.} \lastname{Boutonnet}]{\firstname{Rémi} \lastname{Boutonnet}}
\address{Institut de Mathématiques de Bordeaux, CNRS, Université Bordeaux I\\
33405 Talence, France}
\email{remi.boutonnet@math.u-bordeaux.fr}
\urladdr{https://www.math.u-bordeaux.fr/~rboutonnet/}

\author[\initial{C.} \lastname{Houdayer}]{\firstname{Cyril} \lastname{Houdayer}}
\address{Université Paris-Saclay, Laboratoire de Mathématiques d'Orsay, CNRS, Institut Universitaire de France\\
91405 Orsay, France}
\email{cyril.houdayer@universite-paris-saclay.fr}
\urladdr{https://cyrilhoudayer.com/}

\thanks{RB is supported by ANR grant AODynG 19-CE40-0008. CH is supported by Institut Universitaire de France}

\begin{abstract}
We prove a noncommutative Bader-Shalom factor theorem for lattices with dense projections in product groups. As an application of this result and our previous works, we obtain a noncommutative Margulis factor theorem for all irreducible lattices $\Gamma < G$ in higher rank semisimple algebraic groups. Namely, we give a complete description of all intermediate von Neumann subalgebras $\mathrm{L}(\Gamma) \subset M \subset \mathrm{L}(\Gamma \curvearrowright G/P)$ sitting between the group von Neumann algebra and the group measure space von Neumann algebra associated with the action on the Furstenberg-Poisson boundary.
\end{abstract}

\subjclass{22E40, 37A40, 46L10, 46L55}
\keywords{Factor theorem, Furstenberg-Poisson boundaries, group measure space von Neumann algebras, higher rank lattices in semisimple algebraic groups}

\altkeywords{Théorème du facteur, bords de Furstenberg-Poisson, algèbres de von Neumann d'espace de mesure de groupe, groupes algébriques semi-simples de rang supérieur}

\alttitle{Le théorème du facteur non commutatif pour des réseaux dans des groupes produits}

\begin{altabstract}
Nous montrons un théorème du facteur de Bader-Shalom non commutatif pour les réseaux avec des projections denses dans les groupes produits. Comme application de ce résultat et de nos travaux précédents, nous obtenons un théorème du facteur de Margulis non commutatif pour tous les réseaux irréductibles $\Gamma < G$ dans les groupes algébriques semi-simples de rang supérieur. En particulier, nous donnons une description complète de toutes les sous-algèbres de von Neumann intermédiaires $\mathrm{L}(\Gamma) \subset M \subset \mathrm{L}(\Gamma \curvearrowright G/P)$ situées entre l'algèbre de von Neumann de groupe et l'algèbre de von Neumann d'espace de mesure de groupe associée à l'action sur le bord de Furstenberg-Poisson.
\end{altabstract}
\maketitle
\vspace*{-\baselineskip}
\tableofcontents
\mainmatter

\section{Introduction and statement of the main results}

Let $H$ be a locally compact second countable (lcsc) group. A Borel probability measure $\mu \in \Prob(H)$ is said to be \emph{admissible} if $\mu$ is absolutely continuous with respect to the Haar measure, $\supp(\mu)$ generates $H$ as a semigroup, and $\supp(\mu)$ contains a neighborhood of the identity element $e \in H$. A standard probability space $(X, \nu)$ is said to be a $(H, \mu)$-\emph{space} if it is endowed with a nonsingular action $H \curvearrowright (X, \nu)$ for which the probability measure $\nu$ is $\mu$-\emph{stationary}, that is, $\nu = \mu \ast \nu$.

Following \cite{Fu62a, Fu62b}, for every admissible Borel probability measure $\mu \in \Prob(H)$, we denote by $(B, \nu_B)$ the $(H, \mu)$-\emph{Furstenberg-Poisson boundary}. Recall that $(B, \nu_B)$ is the unique $(H, \mu)$-space for which the $H$-equivariant Poisson transform
\[
\rL^\infty(B, \nu_B) \to \Har^\infty(H, \mu) :\quad f \mto \left( h \mapsto \int_B f(h b) \, {\rm d}\nu_B(b) \right)
\]
is isometric and surjective. Here, $\Har^\infty(H, \mu)$ denotes the space of bounded (right) $\mu$-harmonic functions on $H$. A $(H, \mu)$-\emph{boundary} $(C, \nu_C)$ is a $H$-equivariant measurable factor of $(B, \nu_B)$. In that sense, $(B, \nu_B)$ is the maximal $(H, \mu)$-boundary. For every $(H, \mu)$-boundary $(C, \nu_C)$, we regard $\rL^\infty(C) \subset \rL^\infty(B)$ as a $H$-invariant von Neumann subalgebra. Note that the center $\mathscr Z(H)$ acts trivially on $(B, \nu_B)$. We refer to \cite{Fu00, BS04} for further details on the Furstenberg-Poisson boundary.

\begin{definition}
We say that the pair $(H, \mu)$ satisfies the \emph{boundary freeness condition} if for every nontrivial $(H, \mu)$-boundary $(C, \nu_C)$ and every $h \in H \setminus \mathscr Z(H)$, we have $\nu_C(\Fix_C(h)) = 0$ where $\Fix_C(h) = \left\{c \in C \mid hc = c \right\}$.
\end{definition}

\begin{example}
Let $k$ be a local field, $\bH$ a $k$-isotropic almost $k$-simple linear algebraic $k$-group and $\mu \in \Prob(\bH(k))$ an admissible Borel probability measure. A combination of \cite[Cor.\,5.2]{BS04} and \cite[Lem.\,6.2]{BBHP20} shows that the pair $(\bH(k), \mu)$ satisfies the boundary freeness condition.
\end{example}

Recall that the \emph{quasi-center} $\mathscr{Q}\mathscr{Z}(H)$ is the (not necessarily closed) subgroup of all elements $h \in H$ for which the centralizer $\mathscr Z_H(h)$ is open in $H$. We have $\mathscr Z(H) < \mathscr{Q}\mathscr{Z}(H)$.

In order to state our main result, we introduce the following notation. Let $d \geq 2$. For every $i \in \{1, \dots, d\}$, let $G_i$ be a lcsc group and $\mu_i \in \Prob(G_i)$ an admissible Borel probability measure. Denote by $(B_i, \nu_{B_i})$ the $(G_i, \mu_i)$-Furstenberg-Poisson boundary. Set $(G, \mu) = \prod_{i = 1}^d (G_i, \mu_i)$ and $(B, \nu_B) = \prod_{i = 1}^d (B_i, \nu_{B_i})$. Then $(B, \nu_B)$ is the $(G, \mu)$-Furstenberg-Poisson boundary (see \cite[Cor.\,3.2]{BS04}). For every $i \in \{1, \dots, d\}$, denote by $p_i : G \to G_i$ and by $\widehat p_i : G \to \prod_{j \neq i} G_j$ the canonical homomorphisms.

\begin{definition}
Let $\Gamma < G$ be a lattice, that is, $\Gamma < G$ is a discrete subgroup with finite covolume. We say that $\Gamma < G$ is \emph{embedded with dense projections} if for every $i \in \{1, \dots, d\}$, the restriction $p_i|_\Gamma : \Gamma \to G_i$ is injective and $\widehat p_i(\Gamma) < \prod_{j \neq i} G_j$ is dense.
\end{definition}

Let $\Gamma < G$ be a lattice embedded with dense projections and assume that its center $\mathscr Z(\Gamma)$ is finite. Note that $\mathscr Z(\Gamma) < \prod_{i = 1}^d \mathscr Z(G_i)$ and so $\mathscr Z(\Gamma)$ acts trivially on $(B, \nu_B)$. Set $\Lambda = \Gamma/\mathscr Z(\Gamma)$. Consider the well-defined ergodic action $\Lambda \curvearrowright (B, \nu_B)$ and denote by $\rL(\Lambda \curvearrowright B)$ its associated \emph{group measure space} von Neumann algebra. Whenever $(C, \nu_C)$ is a $(G, \mu)$-boundary, we regard $\rL(\Lambda \curvearrowright C) \subset \rL(\Lambda \curvearrowright B)$ as a von Neumann subalgebra.

Our main result is the following noncommutative analogue of Bader-Shalom's factor theorem for lattices in product groups (see \cite[Th.\,1.7]{BS04}). We call such a result a \emph{Noncommutative Factor Theorem (NFT)}.

\begin{letterthm}[NFT for lattices in products]\label{thmA}
Keep the same notation as above. For every $i \in \{1, \dots, d\}$, assume that $\mathscr{Q}\mathscr{Z}(G_i) = \mathscr Z(G_i)$ and that the pair $(G_i, \mu_i)$ satisfies the boundary freeness condition. Let $\rL(\Lambda) \subset M \subset \rL(\Lambda \curvearrowright B)$ be an intermediate von Neumann subalgebra.

Then for every $i \in \{1, \dots, d\}$, there exists a unique $(G_i, \mu_i)$-boundary $(C_i, \nu_{C_i})$ such that with $(C, \nu_C) = \prod_{i = 1}^d (C_i, \nu_{C_i})$, we have $M = \rL(\Lambda \curvearrowright C)$.
\end{letterthm}

We point out that unlike the proof of the recent NFT for lattices in higher rank simple algebraic $k$-groups obtained in \cite{BBH21} (see also \cite{Ho21}), we cannot rely on the noncommutative Nevo-Zimmer theorem from \cite{BH19, BBH21}. The proof of Theorem \ref{thmA} consists of two steps. Firstly, we show that when $\rL(\Lambda) \subset M \subset \rL(\Lambda \curvearrowright B)$ and $\rL(\Lambda) \neq M$, there exist $i \in \{1, \dots, d\}$ and a nontrivial $(G_i, \mu_i)$-boundary $(C_i, \nu_{C_i})$ such that $\rL(\Lambda \curvearrowright C_i) \subset M$. To do this, we consider the conjugation action $\Lambda \curvearrowright M$, we exploit the assumption that $\mathscr{Q}\mathscr{Z}(G_i) = \mathscr Z(G_i)$ and we use the dichotomy theorem for boundary structures \cite[Th.\,5.8]{BBHP20} (in lieu of the noncommutative Nevo-Zimmer theorem \cite{BH19, BBH21}). Secondly, we exploit the assumption that the pair $(G_i, \mu_i)$ satisfies the boundary freeness condition and we combine Bader-Shalom's factor theorem \cite{BS04} and Suzuki's result \cite{Su18} to show that there exists a unique $(G, \mu)$-boundary $(C, \nu_C)$ such that $M = \rL(\Lambda \curvearrowright C)$.

Next, we apply Theorem \ref{thmA} to the setting of higher rank lattices. We introduce the following notation. Let $d \geq 1$. For every $i \in \{1, \dots, d\}$, let $k_i$ be a local field, $\bG_i$ a simply connected $k_i$-isotropic almost $k_i$-simple linear algebraic $k_i$-group and set $G_i = \bG_i(k_i)$.

\begin{definition}
Set $G = \prod_{i = 1}^d G_i$. We say that $\Gamma < G$ is a \emph{higher rank lattice} if the following conditions are satisfied:
\begin{enumeratei}
\item
$\Gamma < G$ is a discrete subgroup with finite covolume;
\item
If $d \geq 2$, then $\Gamma < G$ is embedded with dense projections;
\item
$\sum_{i = 1}^d \rk_{k_i}(\bG_i) \geq 2$.
\end{enumeratei}
\end{definition}

Observe that the higher rank assumption $\sum_{i = 1}^d \rk_{k_i}(\bG_i) \geq 2$ implies that exactly one of the following two situations happens.
\begin{itemize}
\item Either $d = 1$ (\emph{simple} case). Then $k_1 = k$ and $\bG_1 = \bG$ is an almost $k$-simple algebraic $k$-group such that $\rk_k(\bG) \geq 2$.
\item Or $d \geq 2$ (\emph{semisimple} or \emph{product} case).
\end{itemize}

Let $\Gamma < G$ be a higher rank lattice. For every $i \in \{1, \dots, d\}$, choose a minimal parabolic $k_i$-subgroup $\bP_i < \bG_i$ and set $P_i = \bP_i(k_i)$. Set $P = \prod_{i = 1}^d P_i$ and endow the homogeneous space $G/P$ with its unique $G$-invariant measure class. Note that $\mathscr Z(\Gamma) < \mathscr Z(G) < P$. Set $\Lambda = \Gamma/\mathscr Z(\Gamma)$. Consider the well-defined ergodic action $\Lambda \curvearrowright G/P$ and its associated group measure space von Neumann algebra $\rL(\Lambda \curvearrowright G/P)$.

As an application of Theorem \ref{thmA} and \cite[Th.\,D]{BBH21}, we obtain the following noncommutative analogue of Margulis'\! factor theorem for all higher rank lattices (see \cite[Th.\,IV.2.11]{Ma91}).

\begin{letterthm}[NFT for higher rank lattices]\label{thmB}
Keep the same notation as above. Let $\rL(\Lambda) \subset M \subset \rL(\Lambda \curvearrowright G/P)$ be an intermediate von Neumann subalgebra.

Then for every $i \in \{1, \dots, d\}$, there exists a unique parabolic $k_i$-subgroup $\bP_i < \bQ_i < \bG_i$ such that with $Q = \prod_{i = 1}^d \bQ_i(k_i)$, we have $M = \rL(\Lambda \curvearrowright G/Q)$. In particular, the rank $\sum_{i = 1}^d \rk_{k_i}(\bG_i)$ is an invariant of the inclusion $\rL(\Lambda) \subset \rL(\Lambda \curvearrowright G/P)$.
\end{letterthm}

We actually prove in Theorem \ref{thm-higher-rank} a more general version of Theorem \ref{thmB} where the algebraic $k_i$-group $\bG_i$ may not be simply connected and the lattice $\Gamma < G$ may not be embedded with dense projections. We refer to \cite[\S 6]{BBH21} and \cite[\S 5]{Ho21} for a discussion of the relevance of Theorem \ref{thmB} regarding Connes'\! rigidity conjecture for the group von Neumann algebras of higher rank lattices.

Finally, we apply Theorem \ref{thmA} to the setting of lattices in products of trees. We~introduce the following notation. Let $d\geq 2$. For every $i \in \{1, \dots, d\}$, let $\mathsf T_i$ be a bi-regular tree and denote by $\Aut^+(\mathsf T_i)$ the group of bi-coloring preserving automorphisms of $\mathsf T_i$. Let $\Gamma < \prod_{i = 1}^d \Aut^+(\mathsf T_i)$ be a uniform lattice. Denote by $G_i$ the closure of the image of $\Gamma$ in $\Aut^+(\mathsf T_i)$ and assume that $G_i$ is $2$-transitive on the boundary $\partial \mathsf T_i$. Assume that $\Gamma < \prod_{i = 1}^d G_i$ is embedded with dense projections. Endow $\partial \mathsf T_i$ with its unique $G_i$-invariant measure class and $B = \prod_{i = 1}^d \partial \mathsf T_i$ with the product measure class.

As an application of Theorem \ref{thmA}, we obtain the following noncommutative analogue of Burger-Mozes'\! factor theorem for lattices in product of trees (see \cite[Th.\,4.6]{BM00b}).

\begin{letterthm}[NFT for lattices in products of trees]\label{thmC}
Keep the same notation as above. Let $\rL(\Gamma) \subset M \subset \rL(\Gamma \curvearrowright B)$ be an intermediate von Neumann subalgebra.

Then there exists a unique subset $J \subset \{1, \dots, d\}$ such that with $B_J = \prod_{j \in J} \partial \mathsf T_j$, we have $M = \rL(\Gamma \curvearrowright B_J)$.
\end{letterthm}

\section{Continuous elements in noncommutative boundaries}

Let $H$ be a lcsc group and $M$ a von Neumann algebra. We say that $M$ is a $H$-\emph{von Neumann algebra} if it is endowed with a continuous action $H \curvearrowright M$. Let $M$ and $N$ be $H$-von Neumann algebras. We say that a unital normal mapping $\Psi : M \to N$ is a $H$-\emph{map} if $\Psi$ is $H$-equivariant with respect to the actions $H \curvearrowright M$ and $H \curvearrowright N$.

For $i \in \{1, 2\}$, let $G_i$ be a lcsc group and $\mu_i \in \Prob(G_i)$ an admissible Borel probability measure. Denote by $(B_i, \nu_{B_i})$ the $(G_i, \mu_i)$-Furstenberg-Poisson boundary. Set $(G, \mu) = (G_1 \times G_2, \mu_1 \otimes \mu_2)$ and $(B, \nu_B) = (B_1 \times B_2, \nu_{B_1} \otimes \nu_{B_2})$. Denote by $p_i : G \to G_i$ the canonical homomorphism. Let $\Gamma < G$ be a lattice embedded with dense projections.

Let $M$ be a $\Gamma$-von Neumann algebra endowed with a faithful normal ucp $\Gamma$-map $\Phi : M \to \rL^\infty(B)$. Denote by $\sigma : \Gamma \curvearrowright M$ the action by automorphisms. Following \cite[Def.\,5.3]{BBHP20}, an element $x \in M$ is said to be $G_1$-\emph{continuous} if for every sequence $(\gamma_n)_n$ in $\Gamma$ such that $p_1(\gamma_n) \to e$ in $G_1$, we have $\sigma_{\gamma_n}(x) \to x$ $\ast$-strongly in $M$. The subset $M_1 \subset M$ of all $G_1$-continuous elements in $M$ forms a $\Gamma$-invariant von Neumann subalgebra for which the action $\Gamma \curvearrowright M_1$ extends to a continuous action $G \curvearrowright M$ such that $G_2$ acts trivially (see \cite[Th.\,5.5]{BBHP20}).

Our main result relies on the explicit computation of the subalgebra of continuous elements in the specific context of noncommutative boundaries. More precisely, we regard $\rL(\Gamma \curvearrowright B)$ as a $\Gamma$-von Neumann algebra via the conjugation action $\Gamma \curvearrowright \rL(\Gamma \curvearrowright B)$. The canonical conditional expectation $\rE : \rL(\Gamma \curvearrowright B) \to \rL^\infty(B)$ is a faithful normal ucp $\Gamma$-map. We denote by $u_\gamma \in \rL(\Gamma) \subset \rL(\Gamma \curvearrowright B)$ for $\gamma \in \Gamma$ the unitaries implementing the action $\Gamma \curvearrowright (B, \nu_B)$.

\begin{thm}\label{thm-continuous}
For every $i \in \{1, 2\}$, the von Neumann subalgebra of $G_i$-continuous elements in $\rL(\Gamma \curvearrowright B)$ is equal to $\rL(\Gamma_i \curvearrowright B_i)$, where $\Gamma_i = p_i^{-1}(\mathscr{Q}\mathscr{Z}(G_i)) \cap \Gamma < \Gamma$.
\end{thm}

Note that this theorem extends \cite[Lem.\,5.4]{BBHP20} to the whole group measure space von Neumann algebra $\rL(\Gamma \curvearrowright B)$. The general approach is similar to \cite[Lem.\,5.4]{BBHP20} and relies on the following refinement of Peterson's result \cite[Lem.\,5.1]{Pe14}. We essentially follow his proof, with a little more care to ensure our extra conditions.

\begin{lem}\label{lem-sequence}
As in the statement of Theorem \ref{thm-continuous}, set $\Gamma_1 = p_1^{-1}(\mathscr{Q}\mathscr{Z}(G_1)) \cap \Gamma$.
Let $\gamma \in \Gamma \setminus \Gamma_1$ and $E \subset B_2$ be a nonnull measurable subset. Then there exists a sequence $(\gamma_n)_{n}$ in $\Gamma$ such that $\nu_{B_2}(p_2(\gamma_n) E) \to 1$, $p_1(\gamma_n) \to e$ and $(\gamma_n \gamma \gamma_n^{-1})_{n}$ are pairwise distinct in $\Gamma$.
\end{lem}

\begin{proof}
Set $g = p_1(\gamma) \in G_1\setminus \mathscr{Q}\mathscr{Z}(G_1)$. Then the closed subgroup $\mathscr Z_{G_1}(g)$ has empty interior in $G_1$. Since $G_1$ is a lcsc group, we may choose a compatible proper right invariant metric $d : G_1 \times G_1 \to \R_+$ (see \eg \cite{St73}).

Set $\gamma_0 = e$. We construct by induction a sequence $(\gamma_n)_{n \in \N^*}$ in $\Gamma$ such that for every $n \in \N^*$, the element $\gamma_n \in \Gamma$ satisfies the following three conditions:
\begin{enumeratei}
\item
$1 - \nu_{B_2}(p_2(\gamma_n) E) \leq \sfrac{1}{n}$,
\item
$d(p_1(\gamma_n), e) \leq \sfrac{1}{n}$ and
\item
$p_1(\gamma_{n}) \notin \bigcup_{j = 0}^{n - 1} p_1(\gamma_j)\mathscr Z_{G_1}(g)$.
\end{enumeratei}
Let $n \geq 0$ and assume that we have constructed $\gamma_0, \dots, \gamma_n \in \Gamma$ satisfying the above three conditions. Let us construct $\gamma_{n + 1} \in \Gamma$ that satisfies the above three conditions. Since the action $G_2 \curvearrowright (B_2, \nu_{B_2})$ is nonsingular, we may choose a compact neighborhood $\mathscr K_2 \subset G_2$ of $e \in G_2$ such that
\[
\forall k \in \mathscr K_2, \quad \|{k}_\ast \nu_{B_2} - \nu_{B_2}\|_{1} \leq \frac{1}{2(n + 1)}.
\]
Set $\mathscr F = \bigcup_{j = 0}^n p_1(\gamma_j)\mathscr Z_{G_1}(g)$ and note that $\mathscr F$ is a closed set with empty interior in $G_1$. Consider the open ball $B_{G_1}(e, \sfrac{1}{2(n + 1)})$ of center $e$ and radius $\sfrac{1}{2(n + 1)}$ in~$G_1$. Since $B_{G_1}(e, \sfrac{1}{2(n + 1)}) \cap \mathscr F^c$ is a nonempty open set, we may choose a closed ball $\mathscr C = \overline B_{G_1}(o, r) \subset B_{G_1}(e, \sfrac{1}{2(n + 1)}) \cap \mathscr F^c$ of center $o \in G_1$ and radius $0 < r \leq \sfrac{1}{2(n + 1)}$ in $G_1$. Since $\mathscr C$ is compact and since $\mathscr F$ is closed, we have $\alpha = d(\mathscr C , \mathscr F) > 0$. Note that $\alpha \leq d(\mathscr C, e) \leq \sfrac{1}{2(n + 1)}$. Next, define $\mathscr K_1 = \overline B_{G_1}(e, \sfrac{\alpha}{2})$ to be the closed ball of center $e \in G_1$ and radius $\sfrac{\alpha}{2}$ in $G_1$. For every $y \in \mathscr F$, every $c \in \mathscr C$ and every $k \in \mathscr K_1$, we have
\[
d(kc, y) \geq d(c, y) - d(kc, c) = d(c, y) - d(k, e) \geq \alpha - \frac{\alpha}{2} = \frac{\alpha}{2} > 0.
\]
Thus, we have $\mathscr K_1 \mathscr C \subset \mathscr F^c$. Moreover, for every $k \in \mathscr K_1$ and every $c \in \mathscr C$, we have
\[
d(kc, e) \leq d(kc, c) + d(c, e) = d(k, e) + d(c, e) \leq \frac{\alpha}{2} + \frac{1}{2(n + 1)} \leq \frac{1}{n + 1}.
\]

Denote by $m \in \Prob(G/\Gamma)$ the unique $G$-invariant Borel probability measure. Following the proof of \cite[Lem.\,5.1]{Pe14}, since the pmp action $G_2 \curvearrowright (G/\Gamma, m)$ is ergodic, Kakutani's random ergodic theorem implies that for $\mu_2^{\otimes \N}$-almost every $(\omega_j)_j \in G_2^{\N}$ and $m$-almost every $z\Gamma \in G/\Gamma$, we have
\[
\lim_N \frac{1}{N + 1} \sum_{j = 0}^N \mathbf 1_{(\mathscr K_1 \times \mathscr K_2)\Gamma}(\omega_j^{-1} \cdots \omega_0^{-1} z) = m((\mathscr K_1 \times \mathscr K_2)\Gamma) > 0.
\]
Since $m((\mathscr C \times \mathscr K_2)\Gamma) > 0$, we may choose $z \in \mathscr C \times \mathscr K_2$ such that for $\mu_2^{\otimes \N}$-almost every $(\omega_j)_j \in G_2^{\N}$, the intersection $\{ \omega_j^{-1} \cdots \omega_0^{-1} \mid j \in \N \} \cap (\mathscr K_1 \times \mathscr K_2)\Gamma z^{-1}$ is infinite. Since $(B_2, \nu_{B_2})$ is the $(G_2, \mu_2)$-Furstenberg-Poisson boundary, for $\mu_2^{\otimes \N}$-almost every $\omega = (\omega_j)_j \in G_2^\N$, the limit measure $(\nu_{B_2})_\omega$ is a Dirac mass (see \cite[Th.\,2.14]{BS04}). Moreover, we have $\nu_{B_2} = \int_{G_2^\N} (\nu_{B_2})_\omega\, {\rm d}\mu_2^{\otimes \N}(\omega)$ (see \cite[Th.\,2.10]{BS04}). Therefore, we~have
\[
\mu_2^{\otimes \N} (\{(\omega_j)_j \in G_2^\N \mid \lim_j \nu_{B_2}(\omega_j^{-1} \cdots \omega_0^{-1} p_2(z) E) = 1\}) = \nu_{B_2}(p_2(z) E) > 0.
\]
We may choose $(\omega_j)_j \in G_2^{\N}$ such that the intersection $\{ \omega_j^{-1} \cdots \omega_0^{-1} \mid j \in \N \} \cap (\mathscr K_1 \times\nobreak \mathscr K_2)\Gamma z^{-1}$ is infinite and $\lim_j \nu_{B_2}(\omega_j^{-1} \cdots \omega_0^{-1} p_2(z) E) = 1$. We may then choose $j \in \N$, $h \in \mathscr K_1 \times \mathscr K_2$ and $\gamma_{n + 1} \in \Gamma$ such that $\omega_j^{-1} \cdots \omega_0^{-1} = h \gamma_{n + 1} z^{-1}$ and
\[
1 - \nu_{B_2}(\omega_j^{-1} \cdots \omega_0^{-1} p_2(z) E) \leq \frac{1}{2(n + 1)}.
\]
Then $p_1(\gamma_{n + 1}) = p_1(h^{-1})p_1(z) \in \mathscr K_1 \mathscr C$ and so $p_1(\gamma_{n + 1}) \notin \bigcup_{j = 0}^n p_1(\gamma_j)\mathscr Z_{G_1}(g)$ and $d(p_1(\gamma_{n + 1}), e) \leq \sfrac{1}{n + 1}$. Moreover, we have
\begin{align*}
1 - \nu_{B_2}(p_2(\gamma_{n + 1}) E) &= 1 - \nu_{B_2}(p_2(h^{-1}) \omega_j^{-1} \cdots \omega_0^{-1} p_2(z) E) \\
&= 1 - \nu_{B_2}(\omega_j^{-1} \cdots \omega_0^{-1} p_2(z) E) + \|p_2(h)_\ast \nu_{B_2} - \nu_{B_2}\|_1 \\
&\leq \frac{1}{2(n + 1)} + \frac{1}{2(n + 1)} = \frac{1}{n + 1}.
\end{align*}
Thus, the element $\gamma_{n + 1} \in \Gamma$ satisfies the above three conditions.

The sequence $(\gamma_n)_n$ that we have constructed by induction satisfies the conclusion of the lemma.
\end{proof}

Let $(X, \nu)$ be a standard probability space. For every measurable function \hbox{$f : X \to \C$}, we set
\[
\|f\|_{\nu} = \biggl( \int_X |f(x)|^2 \, {\rm d}\nu(x)\biggr)^{1/2}.
\]

\begin{proof}[Proof of Theorem \ref{thm-continuous}]
We may assume that $i = 1$. Set $\mathscr B = \rL(\Gamma \curvearrowright B)$ and denote by $\mathscr B_1 \subset \mathscr B$ the von Neumann subalgebra of all $G_1$-continuous elements in $\mathscr B$. Set $\varphi = \nu_B \circ \rE \in \mathscr B_\ast$.

Observe that $\rL(\Gamma_1 \curvearrowright B_1) \subset \mathscr B_1$. Indeed, it is obvious that $\rL^\infty(B_1) \subset \mathscr B_1$. Let now $\gamma \in \Gamma_1$ so that $p_1(\gamma) \in \mathscr{Q}\mathscr{Z}(G_1)$. Let $(\gamma_n)_n$ be a sequence in $\Gamma$ such that $p_1(\gamma_n) \to e$. Since $\mathscr Z_{G_1}(p_1(\gamma)) < G_1$ is open, there exists $n_0 \in \N$ such that $p_1(\gamma_n) \in \mathscr Z_{G_1}(p_1(\gamma))$ for every $n \geq n_0$. Since $p_1|_\Gamma : \Gamma \to G_1$ is injective, we have $\gamma_n \in \mathscr Z_\Gamma(\gamma)$ for every $n \geq n_0$. This implies that $u_\gamma \in \mathscr B_1$. Altogether, we obtain $\rL(\Gamma_1 \curvearrowright B_1) \subset \mathscr B_1$.

Next, we prove that $\mathscr B_1 \subset \rL(\Gamma_1 \curvearrowright B_1)$. Firstly, we show that $\mathscr B_1 \subset \rL(\Gamma_1 \curvearrowright B)$. By contraposition, let $x \in \mathscr B \setminus\rL(\Gamma_1 \curvearrowright B)$. Write $x = \sum_{\gamma \in \Gamma} x_\gamma u_\gamma$ for the Fourier expansion of $x \in \mathscr B$ where $x_\gamma = \rE(xu_\gamma^*)$ for every $\gamma \in \Gamma$. Since $x \not\in \rL(\Gamma_1 \curvearrowright B)$, there exists $\gamma \in \Gamma \setminus \Gamma_1$ such that $x_\gamma \neq 0$. We may regard $x_\gamma \in \rL^\infty(B) = \rL^\infty(B_2, \rL^\infty(B_1))$ as a measurable function $f : B_2 \to \rL^\infty(B_1)$. Since $x_\gamma \neq 0$, the measurable function \hbox{$f : B_2 \to \rL^\infty(B_1)$} possesses a nonzero essential value $y\!\in\!\rL^\infty(B_1)$. Set \hbox{$\varepsilon \!=\! \|y\|_{\nu_{B_1}}/2 \!>\! 0$}. Then the measurable subset $E = \{b \in B_2 \mid \|f(b) - y\|_{\nu_{B_1}} < \varepsilon\}$ is nonnull. Choose a sequence $(\gamma_n)_n$ in $\Gamma$ that satisfies the conclusion of Lemma \ref{lem-sequence} for $\gamma \in \Gamma \setminus \Gamma_1$ and $E \subset B_2$. Then
\begin{align*}
\forall n \in \N, \quad \|u_{\gamma_n} x u_{\gamma_n}^* - x\|_\varphi^2 &= \sum_{h \in \Gamma} \|\sigma_{\gamma_n}(x_h) - x_{\gamma_n h \gamma_n^{-1}}\|_{\nu_B}^2 \\
& \geq \|\sigma_{\gamma_n}(x_\gamma) - x_{\gamma_n \gamma \gamma_n^{-1}}\|_{\nu_B}^2.
\end{align*}
On the one hand, since the elements $(\gamma_n \gamma \gamma_n^{-1})_n$ are pairwise distinct in $\Gamma$ and since $\|x\|_\varphi^2 = \sum_{h \in \Gamma} \|x_h\|_{\nu_B}^2 \geq \sum_{n \in \N} \|x_{\gamma_n \gamma \gamma_n^{-1}}\|_{\nu_B}^2$, we have
\[
\lim_n \|x_{\gamma_n \gamma \gamma_n^{-1}}\|_{\nu_B}^2 = 0.
\]
On the other hand, since $p_1(\gamma_n) \to e$, since $\nu_{B_2}(p_2(\gamma_n) E) \to 1$ and since $\|f(b)\|_{\nu_{B_1}} \geq \|y\|_{\nu_{B_1}} - \|f(b) - y\|_{\nu_{B_1}} \geq \varepsilon$ for every $b \in E$, we have
\begin{align*}
\liminf_n\|\sigma_{\gamma_n}(x_\gamma)\|_{ \nu_B}^2 &= \liminf_n \int_{B_2} \|\sigma_{p_1(\gamma_n)}(f(p_2(\gamma_n)^{-1} b))\|^2_{\nu_{B_1}} {\rm d}\nu_{B_2}(b) \\
&= \liminf_n \int_{B_2} \|f(p_2(\gamma_n)^{-1} b)\|^2_{\nu_{B_1} \circ p_1(\gamma_n)} \, {\rm d}\nu_{B_2}(b) \\
&\geq \liminf_n \int_{p_2(\gamma_n) E} \|f(p_2(\gamma_n)^{-1} b)\|^2_{\nu_{B_1} \circ p_1(\gamma_n)} \, {\rm d}\nu_{B_2}(b) \geq \varepsilon^2.
\end{align*}
Altogether, this implies that $\liminf_n \|u_{\gamma_n} x u_{\gamma_n}^* - x\|_\varphi^2 \geq \varepsilon^2$ and so $x \not\in \mathscr B_1$. This shows that $\mathscr B_1 \subset \rL(\Gamma_1 \curvearrowright B)$.

Secondly, we show that $\mathscr B_1 \subset \rL(\Gamma_1 \curvearrowright B_1)$. Indeed, let $x \in \mathscr B_1$. The previous paragraph shows that $x \in \rL(\Gamma_1 \curvearrowright B)$ and so we may write $x = \sum_{\gamma \in \Gamma_1} x_\gamma u_\gamma$, where $x_\gamma = \rE(x u_\gamma^*)$ for every $\gamma \in \Gamma_1$. Since the faithful normal conditional expectation $\rE : \mathscr B \to \rL^\infty(B)$ is $\Gamma$-equivariant, we have that $\rE(\mathscr B_1)$ is contained in the von Neumann subalgebra of $G_1$-continuous elements in $\rL^\infty(B)$, which is equal to $\rL^\infty(B_1)$ by \cite[Lem.\,5.4]{BBHP20}. Since $\rL(\Gamma_1) \subset \mathscr B_1$, we have $x_\gamma = \rE(x u_\gamma^*) \in \rL^\infty(B_1)$ for every $\gamma \in \Gamma_1$. Then \cite[Cor.\,3.4]{Su18} further implies that $\mathscr B_1 \subset \rL(\Gamma_1 \curvearrowright B_1)$. Thus, we have $\mathscr B_1 = \rL(\Gamma_1 \curvearrowright B_1)$.
\end{proof}

Next, we further assume that $\mathscr{Q}\mathscr{Z}(G_i) = \mathscr Z(G_i)$ for every $i \in \{1, 2\}$ and that $\mathscr Z(\Gamma)$ is finite. Set $\Lambda = \Gamma/\mathscr Z(\Gamma)$ and denote by $\pi : \Gamma \to \Lambda$ the quotient homomorphism. Consider the well-defined ergodic action $\Lambda \curvearrowright (B, \nu_B)$. We may regard $\rL(\Lambda \curvearrowright B)$ as a $\Gamma$-von Neumann algebra via the quotient homomorphism $\pi : \Gamma \to \Lambda$ and the conjugation action $\Lambda \curvearrowright \rL(\Lambda \curvearrowright B)$. Moreover, the canonical conditional expectation $\rE : \rL(\Lambda \curvearrowright B) \to \rL^\infty(B)$ is a faithful normal ucp $\Gamma$-map. We derive the following result that will be used in the proof of Theorem \ref{thmA}.

\begin{cor}\label{cor-ergodic}
For every $i \in \{1, 2\}$, the von Neumann subalgebra of $G_i$-continuous elements in $\rL(\Lambda \curvearrowright B)$ is equal to $\rL^\infty(B_i)$. Moreover, the action $\Gamma \curvearrowright \rL(\Lambda \curvearrowright B)$ is ergodic.
\end{cor}

\begin{proof}
The second assertion follows from the first one since a $\Gamma$-invariant element in $\rL(\Lambda \curvearrowright B)$ is both $G_1$-continuous and $G_2$-continuous, hence must be contained in $\rL^\infty(B_1) \cap \rL^\infty(B_2) = \C 1$.

To prove the first assertion, we observe that with the above notation, $\rL(\Lambda \curvearrowright B) \cong z \rL(\Gamma \curvearrowright B)$, where $z = \frac{1}{|\mathscr Z(\Gamma)|} \sum_{h \in \mathscr Z(\Gamma)} u_h \in \mathscr Z(\rL(\Gamma \curvearrowright B))$. Let us be more explicit about this identification. Since $\mathscr Z(\Gamma)$ acts trivially on $B$, the projection $z$ is indeed central in $\rL(\Gamma \curvearrowright B)$. Given $g \in \Gamma$, the element $zu_g$ only depends on $\pi(g) \in \Lambda$. Then the map
\[
\rL(\Lambda \curvearrowright B) \to z\rL(\Gamma \curvearrowright B) : \quad a u_{\pi(g)} \mto zau_g, \quad a \in \rL^\infty(B), g \in \Gamma,
\]
is easily seen to extend to the desired von Neumann algebra isomorphism \[
\Theta : \rL(\Lambda \curvearrowright B) \to z\rL(\Gamma \curvearrowright B).
\]
Note that $\Theta$ is $\Gamma$-equivariant.

Let $i \in \{1,2\}$. Since $\mathscr{Q}\mathscr{Z}(G_i) = \mathscr Z(G_i)$ and since $p_i|_\Gamma : \Gamma \to G_i$ is injective, we have $\Gamma_i = p_i^{-1}( \mathscr Z(G_i)) = \mathscr Z(\Gamma)$. It is obvious that all the elements of $\rL^\infty(B_i)$ are $G_i$\nobreakdash-con\-tinuous in $\rL(\Lambda \curvearrowright B)$. Conversely, let $x \in \rL(\Lambda \curvearrowright B)$ be a $G_i$-continuous element. Then $\Theta(x) \in z\rL(\Gamma \curvearrowright B)$ is $G_i$-continuous in $\rL(\Gamma \curvearrowright B)$. Applying Theorem~\ref{thm-continuous}, we obtain that $\Theta(x) \in \rL(\mathscr Z(\Gamma) \curvearrowright B_i) \cap z\rL(\Gamma \curvearrowright B) = z \rL^\infty(B_i)$. Thus, we have $x \in \rL^\infty(B_i)$. This finishes the proof of the corollary.
\end{proof}

\section{Proofs of the main results}

\begin{proof}[Proof of Theorem \ref{thmA}]
Regard $\mathscr B = \rL(\Lambda \curvearrowright B)$ as a $\Gamma$-von Neumann algebra and denote by $\rE : \mathscr B \to \rL^\infty(B)$ the canonical $\Gamma$-equivariant faithful normal conditional expectation. By Corollary \ref{cor-ergodic}, the action $\Gamma \curvearrowright \mathscr B$ is ergodic. Let $\rL(\Lambda) \subset M \subset \mathscr B$ be an intermediate von Neumann subalgebra. Then $M \subset\mathscr B$ is a $\Gamma$-invariant von Neumann subalgebra and the action $\Gamma \curvearrowright M$ is ergodic. Consider the faithful normal ucp $\Gamma$-map $\Phi = \rE|_M : M \to \rL^\infty(B)$. If $\Phi : M \to \rL^\infty(B)$ is invariant, then for every $x \in M$ and every $\lambda \in \Lambda$, we have $\rE(x u_\lambda^*) \in \C 1$. Then we infer that $M = \rL(\Lambda)$ (see \eg \cite[Cor.\,3.4]{Su18}).

Next, assume that $\Phi : M \to \rL^\infty(B)$ is not invariant. Then \cite[Th.\,5.8]{BBHP20} implies that there exists $i \in \{1, \dots, d\}$ such that the $\Gamma$-invariant von Neumann subalgebra $M_i \subset M$ of all $G_i$-continuous elements in $M$ is nontrivial. Moreover, the action $\Gamma \curvearrowright M_i$ extends to a continuous action $G_i \curvearrowright M_i$ and the ucp map $\Phi|_{M_i} : M_i \to \rL^\infty(B_i)$ is $G_i$-equivariant and not invariant. Since $M \subset \mathscr B$ is $\Gamma$\nobreakdash-in\-vari\-ant, we have $M_i \subset \mathscr B_i$ where $\mathscr B_i \subset \mathscr B$ is the $\Gamma$-invariant von Neumann subalgebra of all $G_i$-continuous elements in $\mathscr B$. Corollary \ref{cor-ergodic} implies that $M_i \subset \rL^\infty(B_i)$. Then $M_i \subset \rL^\infty(B_i)$ is a nontrivial $G_i$-invariant von Neumann subalgebra and so there exists a nontrivial $(G_i, \mu_i)$-boundary $(D_i, \nu_{D_i})$ such that $M_i = \rL^\infty(D_i)$. Then we have $\rL(\Lambda \curvearrowright D_i) \subset M \subset \mathscr B$. Since the pair $(G_i, \mu_i)$ satisfies the boundary freeness condition and since the restriction $p_i |_\Gamma : \Gamma \to G_i$ is injective, it follows that the nonsingular action $\Lambda \curvearrowright (D_i, \nu_{D_i})$ is essentially free. Then a combination of Bader-Shalom's factor theorem \cite[Th.\,1.7]{BS04} and Suzuki's result \cite[Th.\,3.6]{Su18} implies that for every $j \in \{1, \dots, d\}$, there exists a unique $(G_j, \mu_j)$-boundary $(C_j, \nu_j)$ such that with $(C, \nu_C) = \prod_{j = 1}^d (C_j, \nu_{C_j})$, we have $M = \rL(\Lambda \curvearrowright C)$.
\end{proof}

\begin{rem}
We point out that the analogous statement of Theorem \ref{thmA} for \hbox{$\rL(\Gamma \curvearrowright B)$} is false in the case when $\mathscr Z(\Gamma)$ is nontrivial. Indeed, the presence of the central projection $z \in \mathscr Z(\rL(\Gamma \curvearrowright B))$ as defined in the proof of Corollary \ref{cor-ergodic} produces pathological intermediate subalgebras such as $z \rL(\Gamma) \oplus (1-z)\rL(\Gamma \curvearrowright B)$. Nevertheless, our strategy still allows to classify intermediate subalgebras in this~case.
\end{rem}

Next, we prove a general noncommutative factor theorem for higher rank lattices that will imply Theorem \ref{thmB}.

We introduce the following notation. Let $d \geq 1$. For every $i \in \{1, \dots, d\}$, let $k_i$ be a local field and $\bG_i$ a $k_i$-isotropic almost $k_i$-simple linear algebraic $k_i$-group and set $G_i = \bG_i(k_i)$. Denote by $G_i^+ < G_i$ the subgroup generated by the subgroups $\bU(k_i)$ where $\bU$ runs through the set of unipotent $k_i$-split subgroups of $\bG_i$ (see \cite[Prop.\,I.5.4(i)]{Ma91}). Choose a minimal parabolic $k_i$-subgroup $\bP_i < \bG_i$ and set $P_i = \bP_i(k_i)$.
Set $G = \prod_{i = 1}^d G_i$, $G^+ = \prod_{i = 1}^d G_i^+$, $P = \prod_{i = 1}^d P_i$ and endow the homogeneous space $G/P$ with its unique $G$-invariant measure class. Observe that we have $G = G^+ \cdot P$ and so the action $G^+ \curvearrowright G/P$ is transitive (see \cite[Prop.\,I.5.4(vi)]{Ma91}). For every $i \in \{1, \dots, d\}$, denote by $p_i : G \to G_i$ and by $\widehat p_i : G \to \prod_{j \neq i} G_j$ the canonical homomorphisms.

Let $\Gamma < G$ be a lattice such that for every $i \in \{1, \dots, d\}$, we have $\prod_{j \neq i} G_j^+ < \overline{\widehat p_i(\Gamma)}$ and $p_i|_\Gamma : \Gamma \to G_i$ is injective. We point out that $\Gamma < G$ need not be embedded with dense projections. Note that $\mathscr Z(\Gamma) < \mathscr Z(G) < P$ (see \cite[Lem.\,II.6.3(I)]{Ma91}). Set $\Lambda = \Gamma/\mathscr Z(\Gamma)$ and denote by $\pi : \Gamma \to \Lambda$ the quotient homomorphism. Consider the well-defined ergodic action $\Lambda \curvearrowright G/P$ and its associated group measure space von Neumann algebra $\rL(\Lambda \curvearrowright G/P)$. We may regard $\rL(\Lambda \curvearrowright G/P)$ as a $\Gamma$-von Neumann algebra via the quotient homomorphism $\pi : \Gamma \to \Lambda$ and the conjugation action $\Lambda \curvearrowright \rL(\Lambda \curvearrowright G/P)$.

\begin{lem}\label{lem-conjugation}
The action $\Gamma \curvearrowright \rL(\Lambda \curvearrowright G/P)$ is ergodic.
\end{lem}

\begin{proof}
Choose a Borel probability measure $\mu \in \Prob(G)$ of the form $\mu = \psi \cdot m_G$, where $m_G$ is a Haar measure on $G$ and $\psi : G \to \R_+$ is a compactly supported continuous function such that the set $\{g \in G \mid \psi(g) > 0 \text{ and } \psi(g^{-1}) > 0\}$ generates~$G$. Since $G/P$ is compact, there exists a Borel probability measure $\nu \in \Prob(G/P)$ such that $(G/P, \nu)$ is a $(G, \mu)$-space. Moreover, $\nu\in \Prob(G/P)$ belongs to the unique $G$-invariant measure class (see \eg \cite[Lem.\,1.1]{NZ97}).

By \cite[Prop.\,VI.4.1]{Ma91}, there exists a fully supported probability measure $\mu_\Gamma \in \Prob(\Gamma)$ such that $(G/P, \nu)$ is a $(\Gamma, \mu_\Gamma)$-space. Denote by $\mu_\Lambda \in \Prob(\Lambda)$ the pushforward measure of $\mu_\Gamma$ under the quotient homomorphism $\pi : \Gamma \to \Lambda$. Then $(G/P, \nu)$ is a $(\Lambda, \mu_\Lambda)$-space. Since $\Lambda$ has infinite conjugacy classes, \cite[Lem.\,2.6]{KP21} implies that the conjugation action $\Lambda \curvearrowright \rL(\Lambda \curvearrowright G/P)$ is ergodic. Thus, the action $\Gamma \curvearrowright \rL(\Lambda \curvearrowright G/P)$ is ergodic.
\end{proof}

The following theorem is a noncommutative analogue of Margulis'\! factor theorem for all higher rank lattices (see \cite[Th.\,IV.2.11]{Ma91}). It is a more general version of Theorem \ref{thmB}.

\begin{thm}\label{thm-higher-rank}
Keep the same notation as above. Let $\rL(\Lambda) \subset M \subset \rL(\Lambda \curvearrowright G/P)$ be an intermediate von Neumann subalgebra.

Then for every $i \in \{1, \dots, d\}$, there exists a unique parabolic $k_i$-subgroup $\bP_i < \bQ_i < \bG_i$ such that with $Q = \prod_{i = 1}^d \bQ_i(k_i)$, we have $M = \rL(\Lambda \curvearrowright G/Q)$. In particular, the rank $\sum_{i = 1}^d \rk_{k_i}(\bG_i)$ is an invariant of the inclusion $\rL(\Lambda) \subset \rL(\Lambda \curvearrowright G/P)$.
\end{thm}

\begin{proof}
Regard $\mathscr B = \rL(\Lambda \curvearrowright G/P)$ as a $\Gamma$-von Neumann algebra and denote by \hbox{$\rE : \mathscr B \to \rL^\infty(G/P)$} the canonical $\Gamma$-equivariant faithful normal conditional expectation. By Lemma \ref{lem-conjugation}, the action $\Gamma \curvearrowright \mathscr B$ is ergodic. Let $\rL(\Lambda) \subset M \subset \mathscr B$ be an intermediate von Neumann subalgebra and set $\Phi = \rE|_M : M \to \rL^\infty(G/P)$. \hbox{Using} the same notation and following the same argument as in the first paragraph of the proof of Theorem \ref{thmA}, we conclude that if $\Phi : M \to \rL^\infty(G/P)$ is invariant, then $M = \rL(\Lambda)$. Next, assume that $\Phi : M \to \rL^\infty(G/P)$ is not invariant. There are two cases to consider.

Firstly, assume that $d = 1$. Let $k = k_1$ and $\bG = \bG_1$ such that $\rk_k(\bG) \geq 2$. We~proceed as in the proof of \cite[Th.\,D]{BBH21}. By \cite[Th.\,5.4]{BBH21} (see also \cite[Th.\,B]{BH19}), there exist a proper parabolic $k$-subgroup $\bP < \bQ < \bG$ and a $\Gamma$-equivariant unital normal embedding $\iota : \rL^\infty(G/Q) \to M$ such that $\Phi \circ \iota : \rL^\infty(G/Q) \to \rL^\infty(G/P)$ is the canonical embedding with $Q = \bQ(k)$. Then we have $\rL(\Lambda \curvearrowright G/Q) \subset M$. Since $\Lambda \curvearrowright G/Q$ is essentially free (see \cite[Lem.\,6.2]{BBHP20}), a combination of \cite[Th.\,IV.2.11]{Ma91} and \cite[Th.\,3.6]{Su18} shows that there exists a unique parabolic $k$-subgroup $\bQ < \bR < \bG$ such that $M = \rL(\Lambda \curvearrowright G/R)$ with $R = \bR(k)$.

Secondly, assume that $d \geq 2$. Since $\Gamma < G$ need not be embedded with dense projections, we cannot apply Theorem \ref{thmA}. However, we may proceed as in the proof of \cite[Prop.\,6.1]{BBHP20}. Upon permuting the indices, letting $H_1 = \overline{p_1(\Gamma)} < G_1$ and $H_2 = \overline{\widehat p_1(\Gamma)} < \prod_{j = 2}^d G_j$, we have that $\Gamma < H_1 \times H_2$ is a lattice embedded with dense projections and the von Neumann subalgebra $M_1 \subset M$ of all $H_1$-continuous elements in $M$ is nontrivial. Set $B_1 = G_1/P_1$ and $B_2 = \prod_{j = 2}^d G_j/P_j$. Since $G_1^+ < H_1$ and $\prod_{j = 2}^d G_j^+ < H_2$, as explained in the proof of \cite[Prop.\,6.1]{BBHP20} and using \cite[Cor.\,5.2]{BS04}, for every $i \in \{1, 2\}$, we may choose Borel probability measures $\mu_i \in \Prob(H_i)$ and $\nu_i \in \Prob(B_i)$ such that $(B_i, \nu_i)$ is the $(H_i, \mu_i)$-Furstenberg-Poisson boundary. Since $G_1^+ < H_1$ and $\prod_{j = 2}^d G_j^+ < H_2$, using \cite[Th.\,I.1.5.6(i)]{Ma91} and \cite[Prop.\,4.3]{CM08}, we infer that for every $i \in \{1, 2\}$, $\mathscr{Q}\mathscr{Z}(H_i) = \mathscr Z(H_i)$. Moreover, \cite[Lem.\,6.2]{BBHP20} implies that for every $i \in \{1, 2\}$, the pair $(H_i, \mu_i)$ satisfies the boundary freeness condition. We may now apply Theorem \ref{thmA} to obtain the conclusion.

For every $i \in \{1, \dots, d\}$, there are $2^{\rk_{k_i}(\bG_i)}$ intermediate parabolic $k_i$-subgroups $\bP_i < \bQ_i < \bG_i$. The NFT implies that there are $2^{\sum_{i = 1}^d \rk_{k_i}(\bG_i)}$ intermediate von Neumann subalgebras $\rL(\Lambda) \subset M \subset \rL(\Lambda \curvearrowright G/P)$. Thus, the rank $\sum_{i = 1}^d \rk_{k_i}(\bG_i)$ is an invariant of the inclusion $\rL(\Lambda) \subset \rL(\Lambda \curvearrowright G/P)$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thmC}]
For every $i \in \{1, \dots, d\}$, we have $\mathscr{Q}\mathscr{Z}(G_i) = \{e\}$ (see \cite[Lem.\,3.1.1 \& Prop.\,3.1.2]{BM00a}). Moreover, the nonsingular action $\Gamma \curvearrowright \partial \mathsf T_i$ is essentially free (see \cite[Prop.\,6.4]{BBHP20}). This condition is sufficient to apply the proof of Theorem~\ref{thmA} to obtain the conclusion.
\end{proof}

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