The \v{S}ilov 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 \v{S}ilov ideal. The direct proof of Hamana's theorem for the existence of an injective envelope for a unital operator subspace X of some B(H) that we provide implies that the \v{S}ilov ideal is the intersection of C*(X) with any maximal boundary operator subsystem in B(H). As an immediate consequence we deduce that the \v{S}ilov ideal is the biggest boundary operator subsystem for X in C*(X). The new proof of the existence of the \v{S}ilov ideal that we give does not use the existence of maximal dilations, provided by Dritschel and McCullough, and so it is independent of the one given by Arveson. As a countereffect, the \v{S}ilov 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 on \iota(X). The interpretation of our results in terms of ucp maps characterizes the maximal boundary subsystems of X in 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 \v{S}ilov boundary for X.