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

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.