Stable isomorphism of dual operator spaces

Abstract

We prove that two dual operator spaces X and Y are stably isomorphic if and only if there exist completely isometric normal representations ϕ and ψ of X and Y, respectively, and ternary rings of operators M 1 , M 2 such that ϕ ( X ) = [ M 2 ∗ ψ ( Y ) M 1 ] − w ∗ and ψ ( Y ) = [ M 2 ϕ ( X ) M 1 ∗ ] − w ∗ . We prove that this is equivalent to certain canonical dual operator algebras associated with the operator spaces being stably isomorphic. We apply these operator space results to prove that certain dual operator algebras are stably isomorphic if and only if they are isomorphic. Consequently, we obtain that certain complex domains are biholomorphically equivalent if and only if their algebras of bounded analytic functions are Morita equivalent in our sense. Finally, we provide examples motivated by the theory of CSL algebras.