Primitivity of unital full free products of residually finite dimensional C*-algebras

Abstract

A C*-algebra is called primitive if it admits a faithful and irreducible *-representation. We show that if A_1 and A_2 are separable, unital, residually finite dimensional C*-algebras that are not both two dimensional, then their unital C*-algebra full free product, A = A_1*A_2, is primitive. It follows that A is antiliminal and the set of pure states is w*-dense in the state space.