Destabilization

This partly expository paper first supplies the details of a method of factoring a stable C∗C∗-algebra AA as B⊗KB⊗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∗C∗-algebras and a category of “KK-algebras”. We consider this equivalence as “inverting” the stabilization process, that is, a “destabilization”.Furthermore, the method of factoring stable C∗C∗-algebras generalizes to Hilbert bimodules, and an analogous category equivalence between the associated enchilada categories is produced, giving a destabilization for C∗C∗-correspondences.Finally, we make a connection with (double) crossed-product duality.