The Shilov Boundary of an Operator Space - and Applications to the Characterization Theorems and Hilbert C−modules

We study a noncommutative (operator space) version of the ‘boundary’, and in particular the Shilov boundary, of a function space. The main idea is that Hilbert C∗−modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We 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’. We also 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. Date: June 7, 1999. PRESENTED AT THE CANADIAN OPERATOR THEORY AND OPERATOR ALGEBRAS SYMPO-