Duality and operator algebras

Abstract

We investigate some subtle and interesting phenomena in the duality 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 $w^{\ast }$-continuous on dual operator spaces. For example, this yields a new characterization of the $\sigma$-weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and it makes possible a generalization of the theory of $W^{\ast }$-modules to the framework of modules over such algebras. We also give a Banach module characterization of $\sigma$-weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.

19 pages (An error corrected, and results slightly reordered)

References in corpus (3)

• The Calculus of One-Sided $M$-Ideals and Multipliers in Operator Spaces
• Quasimultipliers of Operator Spaces
• Duality and Normal Parts of Operator Modules

Cited by in corpus (2)

• Duality and Normal Parts of Operator Modules
• Duality and Operator Algebras II: Operator Algebras as Banach Algebras