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

TL;DR

This paper investigates conditions under which free products of nuclear C*-algebras admit state-preserving approximations, revealing both positive cases and counterexamples that answer open questions in the field.

Contribution

It establishes criteria for the existence of state-preserving finite-rank maps in homogeneous C*-algebras and provides a counterexample showing such approximations do not always exist.

Findings

Existence of state-preserving approximations under certain faithfulness conditions.

Counterexample with $M_2\otimes C[0,1]$ showing no such approximation exists.

Reduced free products of homogeneous C*-algebras have the completely contractive approximation property.

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 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.