The R-Shilov boundary for a local operator space

TL;DR

This paper explores the existence and properties of the R-Shilov boundary in the context of local operator spaces, extending classical notions of envelopes and boundaries to a locally convex setting.

Contribution

It introduces the concepts of the injective R-envelope and R-C*-envelope for unital local operator spaces, and proves their equivalence with existing constructions by Dosi.

Findings

Established the existence of the R-Shilov boundary ideal in local operator spaces.

Defined the injective R-envelope and R-C*-envelope for unital local operator spaces.

Proved the equivalence of the new injective R-envelope with Dosi's construction.

Abstract

To extend the notion of the injective envelope of a unital operator space to the locally convex case, Dosi (2014) first introduced the notion of the injective R-envelope for a unital operator space and then defined the injective R-envelope for a unital local operator space as the closure of the injective R-envelope for its bounded part. In this paper, we investigate the existence of the Shilov boundary ideal in this context, as defined by Arveson (1969). To do this, by following the conceptual frameworks underlying Hamana's constructions of the injective envelope and the C*-envelope, respectively, we define the notions of the injective R-envelope and the R-C*-envelope for a unital local operator space. Furthermore, we show that the injective R-envelope construction given by us coincides with the one given by Dosi (2014).