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

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

2010-11-10

arxiv.org/abs/1011.2452v1

Mathematics

Operator Algebras

9 pages

Scientific paper

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.

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