Characterizations of ordered operator spaces

TL;DR

This paper provides new abstract characterizations for unital and non-unital operator spaces, focusing on cones of accretive operators and gauges, and explores conditions for complete isometry and order isomorphism.

Contribution

It introduces novel characterizations of operator spaces via accretive operators and gauges, and establishes conditions for their complete isometry and order structure preservation.

Findings

Characterization of unital operator spaces using accretive operators.

Representation of operator spaces with matrix orderings via gauges.

Extension criteria for completely positive maps based on gauges.

Abstract

We demonstrate new abstract characterizations for unital and non-unital operator spaces. We characterize unital operator spaces in terms of the cone of accretive operators (operators whose real part is positive). Defining the gauge of an operator $T \in B(H)$ to be $\|Re(T)_+\|$, we demonstrate an abstract characterization of operator spaces up to complete gauge-isometry. Both of these characterizations preserve the structure of the self-adjoint, positive, and accretive operators, as well as the operator norm. We show that an operator space with a given matrix ordering of positive or accretive cones can be represented completely isometrically and completely order isomorphically if and only if each positive cone is normal, in the sense that $x \leq y \leq z$ implies that $\|y\| \leq \max(\|x\|,\|z\|)$ at each matrix level. This is achieved by showing that normal matrix ordered operator spaces are induced by gauges. We show that inducing gauges are not unique in general. Finally, we show that completely positive completely contractive linear maps on non-unital operator spaces extend to any containing operator system if and only if the operator space is induced by a unique gauge.