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arXiv · 1503.00606

Cuntz-Pimsner Algebras, Crossed Products, and $K$-Theory

Christopher Schafhauser
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Abstract

Suppose $A$ is a $C^*$-algebra and $H$ is a $C^*$-correspondence over $A$. If $H$ is regular in the sense that the left action of $A$ is faithful and is given by compact operators, then we compute the $K$-theory of $\mathcal{O}_A(H) \rtimes \mathbb{T}$ where the action is the usual gauge action. The case where $A$ is an AF-algebra is carefully analyzed. In particular, if $A$ is AF, we show $\mathcal{O}_A(H) \rtimes \mathbb{T}$ is AF. Combining this with Takai duality and an AF-embedding theorem of N. Brown, we show the conditions AF-embeddability, quasidiagonality, and stable finiteness are equivalent for $\mathcal{O}_A(H)$. If $H$ is also assumed to be regular, these finiteness conditions can be characterized in terms of the ordered $K$-theory of $A$.

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Christopher Schafhauser. 2015-03-02. Cuntz-Pimsner Algebras, Crossed Products, and $K$-Theory. https://arxiv.org/abs/1503.00606

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