Stably isomorphic dual operator algebras

Abstract

We prove that two unital dual operator algebras A, B are stably isomorphic if and only if they are Δ-equivalent (Eleftherakis in J Pure Appl Algebra, ArXiv:math. OA/0607489v4, 2007), if and only if they have completely isometric normal representations α,β on Hilbert spaces H, K respectively and there exists a ternary ring of operators \({\mathcal{M} \subset B(H, K)}\) such that \({\alpha (A) = [\mathcal{M}^* \beta(B)\mathcal{M}]^{-w^*}}\) and \({\beta(B) = [\mathcal{M}\alpha(A)\mathcal{M}^*]^{-w^*}.}\)