Applications of Operator Space Theory to Nest Algebra Bimodules

Abstract

Recently, Blecher and Kashyap have generalized the notion of W*-modules over von Neumann algebras to the setting where the operator algebras are σ 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 \({\rm{Alg}(\mathcal M)}\) if and only if there exists a completely isometric normal representation \({\Phi }\) of Y and a nest algebra \({\rm{Alg}(\mathcal N)}\) such that \({\rm{Alg}(\mathcal N) \Phi (Y)\rm{Alg}(\mathcal M)\subset \Phi (Y)}\) while \({\Phi (Y)}\) is implemented by a continuous nest homomorphism from \({\mathcal M}\) onto \({\mathcal N}\) . We describe some properties which are preserved by continuous CSL homomorphisms.