Stable isomorphisms of operator algebras

TL;DR

This paper proves that under certain conditions, stable isomorphisms of operator algebras imply their isometric isomorphisms, with applications to graph algebras and semicrossed products.

Contribution

It establishes that stable isomorphisms of operator algebras with specific diagonal properties imply their isometric isomorphisms, extending previous results.

Findings

Stable isomorphism implies isometric isomorphism for certain operator algebras.

Graph is a complete invariant for various isomorphisms in non-selfadjoint graph algebras.

Results apply to algebras with diagonals satisfying cancellation and specific K-theory groups.

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.