Primitivity of some full group C$^*$-algebras

TL;DR

This paper proves that certain full group C*-algebras, formed from free products of amenable groups, are primitive and possess many faithful irreducible representations, advancing understanding of their structure.

Contribution

It establishes the primitivity and rich representation theory of full group C*-algebras for free products of amenable groups with specific conditions.

Findings

Full group C*-algebra of free product is primitive

Such C*-algebras are often antiliminary

Existence of uncountably many inequivalent faithful irreducible representations

Abstract

We show that the full group C$^*$-algebra of the free product of two nontrivial countable amenable discrete groups, where at least one of them has more than two elements, is primitive. We also show that in many cases, this C$^*$-algebra is antiliminary and has an uncountable family of pairwise inequivalent, faithful irreducible representations.