Ordered involutive operator spaces

TL;DR

This paper constructs the noncommutative Shilov boundary for selfadjoint ordered operator spaces, preserving order in the universal property, and explores maximal and minimal unitizations of these spaces.

Contribution

It introduces a noncommutative Shilov boundary for ordered operator spaces and analyzes their maximal and minimal unitizations, extending previous work in the field.

Findings

Constructed the noncommutative Shilov boundary for ordered operator spaces.

Established the boundary's universal property with order-preserving morphisms.

Analyzed maximal and minimal unitizations of ordered operator spaces.

Abstract

This is a companion to recent papers of the authors; here we construct the `noncommutative Shilov boundary' of a (possibly nonunital) selfadjoint ordered space of Hilbert space operators. The morphisms in the universal property of the boundary preserve order. As an application, we consider `maximal' and `minimal' unitizations of such ordered operator spaces.