Characterizations of Ordered Self-adjoint Operator Spaces

Abstract

We describe how self-adjoint ordered operator spaces, also called non-unital operator systems in the literature, can be understood as $\ast$-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 $\ast$-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.

28 Pages. This is a significant revision of the previous version. The paper has been updated for improved clarity and organization, and new applications have been added

References in corpus (2)

• Spectral truncations in noncommutative geometry and operator systems
• Characterizations of ordered operator spaces

Cited by in corpus (1)

• Characterizations of ordered operator spaces