On free wreath products of classical groups

TL;DR

This paper investigates the properties of generalized free wreath products of classical groups, providing explicit calculations of Haar states and revealing that their associated operator algebras often form full type II_1 factors with simple reduced C*-algebras.

Contribution

It offers explicit Haar state computations and demonstrates that these generalized free wreath products often produce full type II_1 factors and simple C*-algebras with unique traces.

Findings

Von Neumann algebras are often full type II_1 factors.

Reduced C*-algebras are typically simple with unique trace.

Explicit Haar state calculations are provided.

Abstract

We study the generalized free wreath product of classical groups introduced by the first author and Arthur Troupel. We give an explicit computation of the Haar state and deduce important properties of their associated operator algebra: in many cases, the von Neumann algebra is a full type ${\rm II}_1$-factor and the reduced C*-algebra is simple with unique trace.