Operator algebras of free wreath products

TL;DR

This paper characterizes operator algebras of free wreath products, explores their structural and approximation properties, and computes K-theory, revealing non-isomorphism among certain quantum groups.

Contribution

It provides a new description of operator algebras of free wreath products and explicit formulas for invariants, advancing understanding of their structural properties.

Findings

Stability of approximation properties like exactness and Haagerup property.

Structural properties of associated von Neumann algebras such as factoriality and primeness.

Explicit K-theory computations showing non-isomorphism of quantum reflection groups.

Abstract

We give a description of operator algebras of free wreath products in terms of fundamental algebras of graphs of operator algebras as well as an explicit formula for the Haar state. This allows us to deduce stability properties for certain approximation properties such as exactness, Haagerup property, hyperlinearity and K-amenability. We study qualitative properties of the associated von Neumann algebra: factoriality, fullness, primeness and absence of Cartan subalgebra and we give a formula for Connes' $T$-invariant and $\tau$-invariant. We also study maximal amenable von Neumann subalgebras. Finally, we give some explicit computations of K-theory groups for C*-algebras of free wreath products. As an application we show that the reduced C*-algebras of quantum reflection groups are pairwise non-isomorphic.