Rigidity for von Neumann algebras of graph product groups. I. Structure of automorphisms

TL;DR

This paper investigates the rigidity properties of von Neumann algebras associated with specific graph product groups, revealing their automorphism structures using advanced deformation and graph algebra techniques.

Contribution

It provides a detailed description of automorphisms of von Neumann algebras from graph product groups with wreath-like property (T), combining deformation theory with new graph algebra methods.

Findings

All automorphisms of these von Neumann algebras are characterized.

Formulas for automorphisms of reduced C*-algebras are established.

The approach integrates Popa's deformation/rigidity theory with novel graph algebra techniques.

Abstract

In this paper we study various rigidity aspects of the von Neumann algebra $L(\Gamma)$ where $\Gamma$ is a graph product group \cite{Gr90} whose underlying graph is a certain cycle of cliques and the vertex groups are the wreath-like product property (T) groups introduced recently in \cite{CIOS21}. Using an approach that combines methods from Popa's deformation/rigidity theory with new techniques pertaining to graph product algebras, we describe all symmetries of these von Neumann algebras and reduced C$^*$-algebras by establishing formulas in the spirit of Genevois and Martin's results on automorphisms of graph product groups \cite{GM19}.