C*-simplicity of free products with amalgamation and radical classes of groups

TL;DR

This paper characterizes when free products with amalgamation have simple reduced group C*-algebras, introduces a radical class of groups related to simplicity, and provides examples illustrating these properties.

Contribution

It offers new criteria for simplicity of reduced group C*-algebras in amalgamated free products and defines a radical class linking group properties to C*-algebra simplicity.

Findings

A new characterization for simplicity of free product C*-algebras

Existence of an amalgam with trivial kernel but non-simple C*-algebra

Identification of a radical class where simplicity aligns with trivial radical

Abstract

We give new characterizations to ensure that a free product of groups with amalgamation has a simple reduced group C*-algebra, and provide a concrete example of an amalgam with trivial kernel, such that its reduced group C*-algebra has a unique tracial state, but is not simple. Moreover, we show that there is a radical class of groups for which the reduced group C*-algebra of any group is simple precisely when the group has a trivial radical corresponding to this class.