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

In this paper we study various rigidity aspects of the von Neumann algebra $L(\mathrm{\Gamma })$ where $\mathrm{\Gamma }$ 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${}^{\ast }$-algebras by establishing formulas in the spirit of Genevois and Martin’s results on automorphisms of graph product groups \cite{GM19}.