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

TL;DR

This paper computes the $K$-theory of certain Cuntz-Pimsner algebras crossed with the circle group, especially when the base algebra is AF, and explores conditions for AF-embeddability, quasidiagonality, and stable finiteness.

Contribution

It provides explicit $K$-theory calculations for Cuntz-Pimsner algebras with regular correspondences and characterizes finiteness conditions via $K$-theory when the base algebra is AF.

Findings

$K$-theory of $ ext{O}_A(H) times ext{T}$ is computed for regular $H$.

When $A$ is AF, $ ext{O}_A(H) times ext{T}$ is also AF.

Finiteness conditions are equivalent and characterized by $K$-theory for regular $H$.

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 $A$.