A new look at C∗-simplicity and the unique trace property of a group

Abstract

We characterize when the reduced C∗-algebra of a non-trivial group has unique tracial state, respectively, is simple, in terms of Dixmier-type properties of the group C∗-algebra.We also give a simple proof of the recent result by Breuillard, Kalantar, Kennedy and Ozawa that the reduced C∗-algebra of a group has unique tracial state if and only if the amenable radical of the group is trivial.