Cartan subalgebras in von Neumann algebras associated with graph product groups (Q6594736)

Let \(\mathcal M\) be a von Neumann algebra. A von Neumaan subalgebra \(\mathcal A\subseteq \mathcal M\) is said to be a Cartan subalgebra if it is a maximal abelian von Neumann subalgebra (abbreviated MASA) and its normalizer \(\mathcal N_{\mathcal M}(\mathcal A)=\{U\in\mathcal M: U\mbox{ is unitary } U\mathcal AU^*=\mathcal A\}\) generates \(\mathcal M\), as a von Neumann algebra. An interesting area of research is to identify situations when these algebras have no Cartan subalgebras or have a unique Cartan subalgebra, up to unitary conjugacy. The authors consider the problem associated with a crossed-product von Neumann algebra \(L^{\infty}(X)\rtimes \Gamma\) for the graph product group \(\Gamma\). Let \(\mathcal G=(\mathcal V,\mathcal E)\) be a finite, simple (no self-loops or multiple edges) graph, where \(\mathcal V\) and \(\mathcal E\) denote its vertices and edges sets, respectively. Let \(\{\Gamma_v\}_{v\in\mathcal V}\) be a family of groups called vertex groups. The graph product group associated with this data, denoted by \(\Gamma =\mathcal G\{\Gamma_v\}\), is the group generated by \(\Gamma_v\), \(v\in\mathcal V\) subject to the relations of the groups \(\Gamma_v\) along with the relations \([\Gamma_u,\Gamma_v]=1\), whenever \((u,v)\in \mathcal E\). For any measure-preserving action \(\Gamma \curvearrowright X\) of \(\Gamma\) on a probability space \(X\), denoted by \(L^{\infty}(X)\rtimes \Gamma\) the crossed-product von Neumann algebra. If \(X\) consists of a singleton, then this amounts to the group von Neumann algebra \(\mathcal L(\Gamma)\). First, they give the following well-known result.\N\NTheorem 1. Let \(\Gamma =\mathcal G\{\Gamma_v\}\) be an icc graph product of groups where \(\mathcal G\) does not admit a de Rham join decomposition of the form \(\mathcal G=\mathcal G_1\circ\mathcal G_2\circ\cdots \circ \mathcal G_k\) such that for all \(2\leq i\leq k\), we have \(\mathcal G_i=(\{v_i,w_i\},\emptyset)\) and \(\Gamma_{v_i}=\Gamma_{w_i}\cong\mathbb Z_2\). Then \(\mathcal L(\Gamma)\) has no Cartan subalgebra.\N\NIn addition, certain rigidity for graph products groups are also considered.\N\NTheorem 2. Let \(\Gamma =\mathcal G\{\Gamma_v\}\) be an graph product of groups such that the graph \(\mathcal G\) does not admit a non-trivial join decomposition \( \mathcal G=\mathcal H\circ \mathcal K\), where \(\mathcal H=(\{v\},\emptyset)\) or \(\mathcal H=(\{v,w\},\emptyset)\) with \(\Gamma_{v}\cong\Gamma_{w}\cong\mathbb Z_2\). Let \(\Gamma \curvearrowright X\) be any free ergodic pmp action, and let \(\mathcal M=L^{\infty}(X)\rtimes \Gamma\) be the corresponding group measure space von Neumann algebra. Then for any Cartan subalgebra \(\mathcal A\subseteq \mathcal M\), there is a unitary perator \(U\in\mathcal M\) such that \(\mathcal A=UL^{\infty}(X)U^*\).