Cuntz–Pimsner algebras, crossed products, and K-theory

Suppose A   is a C⁎C⁎-algebra and H   is a C⁎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 OA(H)⋊TOA(H)⋊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 OA(H)⋊TOA(H)⋊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 OA(H)OA(H). If H is also assumed to be regular, these finiteness conditions can be characterized in terms of the ordered K-theory of A.