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

Suppose $A$ is a $C^{\ast }$-algebra and $H$ is a $C^{\ast }$-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 $$\mathcal{O}$_A(H) \rtimes $\mathbb{T}$$wheretheactionistheusualgaugeaction\mathrm{.}Thecasewhere$A$isanAF-algebraiscarefullyanalyzed\mathrm{.}Inparticular,if$A$ is AF, we show ${\mathcal{O}}_A(H)⋊\mathbb{T}$ 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 $A$.