The Shilov Boundary of an Operator Space and the Characterization Theorems

Abstract

We study operator spaces, operator algebras, and operator modules from the point of view of the noncommutative Shilov boundary. In this attempt to utilize some noncommutative Choquet theory, we find that Hilbert C*-modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We introduce certain multiplier operator algebras and C*-algebras of an operator space, which generalize the algebras of adjointable operators on a C*-module and the imprimitivity C*-algebra. It also generalizes a classical Banach space notion. This multiplier algebra plays a key role here. As applications of this perspective, we unify and strengthen several theorems characterizing operator algebras and modules. We also include some general notes on the commutative case of some of the topics we discuss, coming in part from joint work with Christian Le Merdy, about function modules.