Destabilization

TL;DR

This paper develops a categorical framework for destabilizing stable C*-algebras and C*-correspondences, providing a canonical factorization method and establishing equivalences that invert the stabilization process.

Contribution

It introduces a categorical destabilization method for stable C*-algebras and C*-correspondences, generalizing existing dualities and duality frameworks.

Findings

Categorical equivalence between nondegenerate C*-algebras and K-algebras.

Canonical factorization method for stable C*-algebras.

Extension of destabilization to Hilbert bimodules and C*-correspondences.

Abstract

This partly expository paper first supplies the details of a method of factoring a stable C*-algebra A as B \otimes K in a canonical way. Then it is shown that this method can be put into a categorical framework, much like the crossed-product dualities, and that stabilization gives rise to an equivalence between the nondegenerate category of C*-algebras and a category of "K-algebras". We consider this equivalence as "inverting" the stabilization process, that is, a "destabilization". Furthermore, the method of factoring stable C*-algebras generalizes to Hilbert bimodules, and an analogous category equivalence between the associated enchilada categories is produced, giving a destabilization for C*-correspondences. Finally, we make a connection with (double) crossed-product duality.