Characterizations of Ordered Self-adjoint Operator Spaces

TL;DR

This paper introduces a new perspective on self-adjoint ordered operator spaces using matrix gauge structures, revealing advantages like the inclusion of injective objects and a duality theorem, with applications to subspace characterization and dual space structures.

Contribution

It presents a novel framework for non-unital operator systems via matrix gauge $*$-vector spaces, enabling new duality results and characterizations.

Findings

Inclusion of injective objects in the category of matrix gauge $*$-vector spaces

A Webster-Winkler-type duality theorem for these spaces

Characterization of subspaces as kernels of completely positive maps

Abstract

We describe how self-adjoint ordered operator spaces, also called non-unital operator systems in the literature, can be understood as $*$-vector spaces equipped with a matrix gauge structure. We explain how this perspective has several advantages over other notions of non-unital operator systems in the literature. In particular, the category of matrix gauge $*$-vector spaces includes injective objects and a Webster-Winkler-type duality theorem, both of which we show generally fail with other notions of non-unital operator systems. As applications, we characterize those subspaces of operator systems which are kernels of completely positive maps and define a new operator space structure on the matrix ordered dual of an operator system generalizing the classical notion of a base norm space.