The Shilov Boundary of an Operator Space and the Characterization Theorems

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, in a way that seems to give more information than other current proofs. 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’. THE MAJOR RESULTS IN THIS PAPER WERE PRESENTED AT THE CANADIAN OPERATOR THEORY AND OPERATOR ALGEBRAS SYMPOSIUM, May 2