$\mathcal{O}_{\infty}$ and $\mathcal{Z}$ stable isomorphism as equivalence

Question

If $O$ is a (strongly ?) self absorbing $C^*$-algebra, one has an equivalence relation on (separable) $C^*$-algebras: "being $O$-stably isomorphic" i.e. $A$ and $B$ are $O$-stably isomorphic if and only if there is an isomorphism $A \otimes O \simeq B \otimes O$.

In the case where $O$ is the algebra of compact operator, (and if we restrict to separable algebra) this relation is better understood as Morita equivalence, i.e. equivalence of the category of Hilbert modules, or isomorphisms in the category of (proper) correspondences.

My question is: for other self absorbing algebra, and more precisely for the Cuntz algebra $\mathcal{O}_{\infty}$ and for the Jiang-su algebra $\mathcal{Z}$, is there something comparable to Morita equivalences to have a better (more categorical if you prefer) understanding of stable isomorphisms. For example, is there something that plays the role of the category of Hilbert modules that one can attach to any $C^*$-algebra and that classifies them up to $\mathcal{Z}$-stable isomorphism. Or maybe some sort of category of correspondences ?

Answer 1

Maybe the following can count if you are willing to restrict to separable, simple, stable, nuclear $C^*$-algebras $A$ and $B$. Then the Kirchberg-Phillips Classification Theorem says that $A\otimes\mathcal O_\infty\cong B\otimes\mathcal O_\infty$ if and only if $A$ and $B$ are KK-equivalent, that is, isomorphic in the KK-theory category. KK-theory is defined in terms of certain bimodules, so there is some analogy to correspondences.

(One can get rid of the simplicity assumption above by asking $A$ and $B$ to have isomorphic ideal lattices and being equivalent in a version of KK-theory that keeps track of the ideals, see here.)