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 uti lize some noncommutative Choquet theory. we find that Hilbert C*-modules an d their properties, which Me studied earlier in the operator space framewor k, replace certain topological tools. We introduce certain multiplier opera tor algebras and C*-algebras of an operator space, which generalize the alg ebras of adjointable operators on a C*-module and the imprimitivity C*-alge bra. It also generalizes a classical Banach space notion. This multiplier a lgebra plays a key role here. As applications of this perspective. we unify and strengthen several theorems characterizing operator algebras and modul es. We also include some general notes on the commutative case of some of t he topics we discuss, coming in part from joint work with Christian Le Merd y, about function modules. (C) 2001 Academic Press.