The Šilov Boundary for Operator Spaces

Abstract

Motivated by the recent interest in the examination of unital completely positive maps and their effects in C*-theory, we revisit an older result concerning the existence of the Šilov ideal. The direct proof of Hamana’s Theorem for the existence of an injective envelope for a unital operator subspace X of some \({\mathcal{B}(H)}\) that we provide implies that the Šilov ideal is the intersection of C*(X) with any maximal boundary operator subsystem in \({\mathcal{B}(H)}\) . As an immediate consequence we deduce that the Šilov ideal is the biggest boundary operator subsystem for X in C*(X). The new proof of the existence of the Šilov ideal that we give does not use the existence of maximal dilations, provided by Drits- chel and McCullough, and so it is independent of the one given by Arveson. As a consequence, the Šilov ideal can be seen as the set that contains the abnormalities in a C*-cover \({(C, \iota)}\) of X for all the extensions of the identity map \({{\rm id}_{\iota(X)}}\) . The interpretation of our results in terms of ucp maps characterizes the maximal boundary subsystems of X in \({\mathcal{B}(H)}\) as kernels of X-projections that induce completely minimal X-seminorms; equivalently, X-minimal projections with range being an injective envelope, that we view from now on as the Šilov boundary for X.