Paper detail

Free Products and the Lack of State Preserving Approximations of Nuclear C*-algebras

Let $A$ be a homogeneous C*-algebra and $ϕ$ a state on $A.$ We show that if $ϕ$ satisfies a certain faithfulness condition, then there is a net of finite-rank, unital completely positive, $ϕ$-preserving maps on $A$ that tend to the identity pointwise. This combined with results of Ricard and Xu show that the reduced free product of homogeneous C*-algebras with respect to these states have 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.