Stable isomorphisms of operator algebras

Let $\text{A}$ and $\text{B}$ be operator algebras with $c_0$-isomorphic diagonals and let $\text{K}$ denote the compact operators. We show that if $\text{A}\otimes \text{K}$ and $\text{B}\otimes \text{K}$ are isometrically isomorphic, then $\text{A}$ and $\text{B}$ are isometrically isomorphic. If the algebras $\text{A}$ and $\text{B}$ satisfy an extra analyticity condition a similar result holds with $\text{K}$ 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 $\text{bbZ}$. This has implications in the study of stable isomorphisms between various semicrossed products.