  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.
