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.