Free products and the lack of state-preserving approximations of nuclear $C^*$-algebras

Abstract:

Let $A$ be a homogeneous $C^*$-algebra and $\phi$ a state on $A.$ We show that if $\phi$ satisfies a certain faithfulness condition, then there is a net of finite-rank, unital completely positive, $\phi$-preserving maps on $A$ that tend to the identity pointwise. This, combined with results of Ricard and Xu, shows that the reduced free product of homogeneous $C^*$-algebras with respect to these states has the completely contractive approximation property. We also give an example of a faithful state on $M_2\otimes C[0,1]$ for which no such state-preserving approximation of the identity map exists, thus answering a question of Ricard and Xu.