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

TL;DR

This paper proves that the unital full free product of certain residually finite dimensional C*-algebras is primitive, implying it admits a faithful irreducible *-representation and has a dense set of pure states.

Contribution

It establishes the primitivity of unital full free products of separable, residually finite dimensional C*-algebras under specific conditions, extending understanding of their representation theory.

Findings

Full free products are primitive unless both algebras are two-dimensional.

The resulting algebra is antiliminal with a dense set of pure states.

Primitivity holds for a broad class 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.