Applications of operator space theory to nest algebra bimodules

Recently Blecher and Kashyap have generalized the notion of W* modules over von Neumann algebras to the setting where the operator algebras are \sigma- weakly closed algebras of operators on a Hilbert space. They call these modules weak* rigged modules. We characterize the weak* rigged modules over nest algebras . We prove that Y is a right weak* rigged module over a nest algebra Alg(M) if and only if there exists a completely isometric normal representation \phi of Y and a nest algebra Alg(N) such that Alg(N)\phi(Y)Alg(M) \subset \phi(Y) while \phi(Y) is implemented by a continuous nest homomorphism from M onto N. We describe some properties which are preserved by continuous CSL homomorphisms

Morita embeddings for dual operator algebras and dual operator spaces

We define a relation < for dual operator algebras. We say that B < A if there exists a projection p in A such that B and pAp are Morita equivalent in our sense. We show that < is transitive, and we investigate the following question: If A < B and B < A, then is it true that A and B are stably isomorphic? We propose an analogous relation < for dual operator spaces, and we present some properties of < in this case

Stable isomorphism and strong Morita equivalence of operator algebras

We introduce a Morita type equivalence: two operator algebras $A$ and $B$ are called strongly $\Delta$-equivalent if they have completely isometric representations $\alpha$ and $\beta$ respectively and there exists a ternary ring of operators $M$ such that $\alpha (A)$ (resp. $\beta (B)$) is equal to the norm closure of the linear span of the set $M^*\beta (B)M,$ (resp. $M\alpha (A)M^*$). We study the properties of this equivalence. We prove that if two operator algebras $A$ and $B,$ possessing countable approximate identities, are strongly $\Delta$-equivalent, then the operator algebras A\otimes \cl K and B\otimes \cl K are isomorphic. Here \cl K is the set of compact operators on an infinite dimensional separable Hilbert space and $\otimes$ is the spatial tensor product. Conversely, if A\otimes \cl K and B\otimes \cl K are isomorphic and $A, B$ possess contractive approximate identities then $A$ and $B$ are strongly $\Delta$-equivalent.Comment: We present some shorter proofs using references from the literature. Also example 3.7 is new and provides a new proof of the fact our notion of strong Morita equivalence is stronger than "BMP strong Morita equivalence " introduced by Blecher, Muhly and Paulse

A Morita Type Equivalence for Dual Operator Algebras

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 and B have completely isometric normal representations alpha, beta such that alpha(A) is the w*-closed span of M*beta(B)M and beta(B) is the w*-closed span of Malpha(A)M* for a ternary ring of operators M (i.e. a linear space M such that MM*M \subset M if and only if there exists an equivalence functor $F:_{A}M\to_{B}M$ 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 $\Delta (A)$-module maps as morphisms ($\Delta (A)=A\cap A^*$). We prove that this functor is equivalent to a functor "generated" by a B, A bimodule, that it is normal and completely isometric

Ranges of bimodule projections and reflexivity

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