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 operatorsWe 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.

Morita embeddings for dual operator algebras and dual operator spaces

“…Y * (K 2 )), then the spaces p 2 X| p 1 (H 1 ) , q 2 Y | q 1 (K 1 ) are nondegenerate, and also weakly TRO-equivalent. This can be concluded from Proposition 2.2 in [13].…”

“…An important property is that two algebras are ∆-equivalent if and only if they are stably isomorphic, as was proved by Paulsen and the present author in [12]. Subsequently, Paulsen, Todorov and the present author defined a Morita-type equivalence ∼ ∆ for dual operator spaces [13]. This equivalence also has the property of being equivalent with the notion of a stable isomorphism.…”

“…where α : A → α(A), ζ : Z → ζ(Z), β : B → β(B) are completely isometric maps. By Lemma 2.8 in [13], the TRO N is of the form N = N 2 ⊕ N 1 . Thus,…”

Strong Morita equivalence of operator spaces

“…The appropriate Morita Theorem IV in this setting was later given by the first author and Paulsen [18]. A further generalization to the broader class of dual operator spaces was achieved by the first author with Paulsen and Todorov [19]. Both extensions of Morita theory (versions (1.1) and (1.2)) have advantages and a number of applications, e.g.…”

Stable Properties of Hyperrelexivity

“…Recently a new equivalence relation between weak* closed operator spaces acting on Hilbert spaces has appeared: Definition 1.1. [7] Let H i , K i , i = 1, 2 be Hilbert spaces, and U ⊂ B(K 1 , K 2 ), V ⊂ B(H 1 , H 2 ) be weak* closed spaces. We call them weak TRO equivalent if there exist ternary rings of operators (TRO's) M i ⊂ B(H i , K i ), i = 1, 2, i.e.…”