核C* 代数的自由积与缺乏保态逼近
Free Products and the Lack of State Preserving Approximations of Nuclear C*-algebras
摘要：
设$A$是一个齐次C* 代数，$ϕ$是$A.$上的一个状态.我们证明了如果$ϕ$满足一定的忠实性条件，则在$A$上存在一个由有限秩、单位完全正的、$ϕ$ -保持映射组成的网，该网逐点趋于恒等.结合Ricard和Xu的结果，证明了齐次C* 代数关于这些态的约化自由积具有完全压缩逼近性质.我们还给出了一个例子，一个忠实的状态$M_2\otimes C[0,1]$，没有这样的状态保持近似的身份映射存在，从而回答了一个问题的Ricard和徐。
英文摘要：
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.
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