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Abstract Let $${{\mathcal{A}}}$$ and $${{\mathcal{B}}}$$ be operator algebras with $$c_{0}$$-isomorphic diagonals and let $${{\mathcal{K}}}$$ denote the compact operators. We show that if $${{\mathcal{A}}}\otimes{{\mathcal{K}}}$$ and $${{\mathcal{B}}}\otimes{{\mathcal{K}}}$$ are isometrically isomorphic, then $${{\mathcal{A}}}$$ and $${{\mathcal{B}}}$$ are isometrically isomorphic. If the algebras $${{\mathcal{A}}}$$ and $${{\mathcal{B}}}$$ satisfy an extra analyticity condition a similar result holds with $${{\mathcal{K}}}$$ being replaced by any operator algebra containing the compact operators. For nonselfadjoint 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 $${{\mathbb{Z}}}$$. This has implications in the study of stable isomorphisms between various semicrossed products.