We define an equivalence relation between bimodules over maximal abelian selfadjoint algebras (masa bimodules) which we call spatial Morita equivalence. We prove that two reflexive masa bimodules are spatially Morita equivalent iff their (essential)
bilattices are isomorphic. We also prove that if S^1, S^2 are bilattices which correspond to reflexive masa bimodules U_1, U_2 and f: S^1rightarrow S^2 is an onto bilattice homomorphism, then: (i) If U_1 is synthetic, then U_2 is synthetic. (ii) If U_2 contains a nonzero compact (or a finite or a rank 1) operator, then U_1 also contains a nonzero compact (or a finite or a rank 1) operator.
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 $M$ suc
h that $alpha (A)$ (resp. $beta (B)$) is equal to the norm closure of the linear span of the set $M^*beta (B)M, $ (resp. $Malpha (A)M^*$). 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 $Aotimes cl K$ and $Botimes cl K$ 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 $Aotimes cl K$ and $Botimes cl K$ are isomorphic and $A, B$ possess contractive approximate identities then $A$ and $B$ are strongly $Delta $-equivalent.
Recently a new equivalence relation between weak* closed operator spaces acting on Hilbert spaces has appeared. Two weak* closed operator spaces U, V are called weak TRO equivalent if there exist ternary rings of operators M_i, i=1,2 such that U=[ M_
2 V M_1^*]^{-w^*}, V=[ M_2^* U M_1]^{-w^*} . Weak TRO equivalent spaces are stably isomorphic, and conversely, stably isomorphic dual operator spaces have normal completely isometric representations with weak TRO equivalent images. In this paper, we prove that if cl U and V are weak TRO equivalent operator spaces and the space of I x I matrices with entries in U, M_I^w( U), is hyperreflexive for suitable infinite I, then so is M_I^w( V). We describe situations where if L1, L are isomorphic lattices, then the corresponding algebras Alg{L1}, Alg{L2} have the same complete hyperreflexivity constant.
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 module
s. 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.
