Abstract. We prove that two unital dual operator algebras A, B are stably isomorphic if and only if they are ∆-equivalent [7], 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 M ⊂ B(H, K) such that α(A) = [M * β(B)M]

We generalize the main theorem of Rieffel for Morita equivalence of W * -algebras to the case of unital dual operator algebras: two unital dual operator algebras A, B have completely isometric normal representations α, β such that α(A) = [M * β(B)M] −w * and β(B) = [Mα(A)M * ] −w * for a ternary ring of operators M (i.e. a linear space M such that MM * M ⊂ M) if and only if there exists an equivalence functor F : A M → B M which "extends" to a * -functor implementing an equivalence between the categories A DM and B DM. By A M we denote the category of normal representations of A and by A DM the category with the same objects as A M and ∆(A)-module maps as morphisms (∆(A) = A ∩ A * ). We prove that this functor is equivalent to a functor "generated" by a B, A bimodule, and that it is normal and completely isometric.

We develop a general framework for reflexivity in dual Banach spaces, motivated by the question of when the weak* closed linear span of two reflexive masa-bimodules is automatically reflexive. We establish an affirmative answer to this question in a number of cases by examining two new classes of masa-bimodules, defined in terms of ranges of masa-bimodule projections. We give a number of corollaries of our results concerning operator and spectral synthesis, and show that the classes of masa-bimodules we study are operator synthetic if and only if they are strong operator Ditkin.

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.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International licence Newcastle University ePrints-eprint.ncl.ac.uk Eleftherakis GK, Kakariadis ETA. Strong Morita equivalence of operator spaces.