Stable isomorphisms of operator algebras

Abstract: Let $\A$ and $\B$ be operator algebras with $c_0$-isomorphic diagonals and let $\K$ denote the compact operators. We show that if $\A\otimes\K$ and $\B\otimes\K$ are isometrically isomorphic, then $\A$ and $\B$ are isometrically isomorphic. If the algebras $\A$ and $\B$ satisfy an extra analyticity condition a similar result holds with $\K$ being replaced by any operator algebra containing the compact operators. For non-selfadjoint graph algebras this implies that the graph is a complete invariant for various types of isomorphisms, including stable isomorphisms, thus strengthening a recent result of Dor-On, Eilers and Geffen. Similar results are proven for algebras whose diagonals satisfy cancellation and have $K_0$-groups isomorphic to $\bbZ$. This has implications in the study of stable isomorphisms between various semicrossed products.