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

A C⁎C⁎-algebra is called primitive if it admits a faithful and irreducible ⁎-representation. We show that if A1A1 and A2A2 are separable, unital, residually finite dimensional C⁎C⁎-algebras satisfying (dim⁡(A1)−1)(dim⁡(A2)−1)≥2(dim⁡(A1)−1)(dim⁡(A2)−1)≥2, then the unital C⁎C⁎-algebra full free product, A=A1⁎A2A=A1⁎A2, is primitive. It follows that A   is antiliminal, it has an uncountable family of pairwise inequivalent irreducible faithful ⁎-representations and the set of pure states is w⁎w⁎-dense in the state space.