Duality and Operator Algebras

We investigate some subtle and interesting phenomena in the du-ality theory of operator spaces and operator algebras. In particular, we give several applications of operator space theory, based on the surprising fact that certain maps are always weak *-continuous on dual operator spaces. For example , if X is a subspace of a C *-algebra A, and if a ∈ A satisfies aX ⊂ X and a * X ⊂ X, and if X is isometric to a dual Banach space, then we show that the function x → ax on X is weak * continuous. Applications include a new characterization of the σ-weakly closed (possibly nonunital and nonselfad-joint) operator algebras, and it makes possible a generalization of the theory of W *-modules to the framework of modules over such algebras. We also give a Banach module characterization of σ-weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.