Paper detail

The Šilov Boundary for Operator Spaces

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 B(H) that we provide implies that the Šilov ideal is the intersection of C*(X) with any maximal boundary operator subsystem in 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 Dritschel and McCullough, and so it is independent of the one given by Arveson. As a countereffect, the Šilov ideal can be seen as the set that contains the abnormalities in a C*-cover (C,ι) of X for all the extensions of the identity map on ι(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 Šilov boundary for X.