The \v{S}ilov Boundary for Operator Spaces

TL;DR

This paper provides a new, independent proof of the existence and characterization of the ilov boundary for operator spaces, linking it to boundary operator subsystems and injective envelopes without relying on maximal dilations.

Contribution

It offers a novel proof of the ilov ideal's existence and properties, independent of previous dilation-based approaches, and characterizes the boundary in terms of ucp maps and minimal projections.

Findings

ilov ideal is the intersection of C*(X) with maximal boundary operator subsystems.

The ilov ideal is the largest boundary operator subsystem for X.

The new proof does not depend on maximal dilations, providing an alternative perspective.

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.