References & Citations
Mathematics > Operator Algebras
Title: Rigidity results for group von Neumann algebras with diffuse center
(Submitted on 2 Mar 2024 (v1), last revised 27 Mar 2024 (this version, v2))
Abstract: We introduce the first examples of groups $G$ with infinite center which in a natural sense are completely recognizable from their von Neumann algebras, $\mathcal{L}(G)$. Specifically, assume that $G=A\times W$, where $A$ is an infinite abelian group and $W$ is an ICC wreath-like product group [CIOS22a; AMCOS23] with property (T) and trivial abelianization. Then whenever $H$ is an \emph{arbitrary} group such that $\mathcal{L}(G)$ is $\ast$-isomorphic to $\mathcal L(H)$, via an \emph{arbitrary} $\ast$-isomorphism preserving the canonical traces, it must be the case that $H= B \times H_0$ where $B$ is infinite abelian and $H_0$ is isomorphic to $W$. Moreover, we completely describe the $\ast$-isomorphism between $\mathcal L(G)$ and $\mathcal L(H)$. This yields new applications to the classification of group C$^*$-algebras, including examples of non-amenable groups which are recoverable from their reduced C$^*$-algebras but not from their von Neumann algebras.
Submission history
From: Adriana Fernández Quero [view email][v1] Sat, 2 Mar 2024 18:25:30 GMT (91kb)
[v2] Wed, 27 Mar 2024 03:06:18 GMT (97kb)
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