Quantization of $A_{0}(K)$-Spaces

TL;DR

This paper characterizes C*-ordered operator spaces via $A_{0}(K)$-spaces derived from $L^1$-matrix convex sets, extending the theory of regular embeddings to operator systems.

Contribution

It introduces the concept of $L^{1}$-regular embedding for $L^{1}$-matrix convex sets and characterizes C*-ordered operator spaces as $A_{0}(K)$-spaces.

Findings

C*-ordered operator spaces are completely isometrically isomorphic to $A_{0}(K)$-spaces.

Conditions are established for $A_{0}(K)$-spaces to be abstract operator systems.

Generalization of regular embedding to $L^{1}$-regular embedding for $L^{1}$-matrix convex sets.

Abstract

In this paper, we study $L^1$-matrix convex sets $\{K_{n}\}$ in $*$-locally convex spaces and show that every C$^*$-ordered operator space is complete isometrically, completely isomorphic to $\{A_{0}(K_{n}, M_{n}(V))\}$ for a suitable $L^1$-matrix convex set $\{K_{n}\}$. Further, we generalize the notion of regular embedding of a compact convex set to $L^{1}$-regular embedding of $L^{1}$-matrix convex set. Using $L^{1}$-regular embedding of $L^{1}$-convex set, we find conditions under which $A_{0}(K_{n}, M_{n}(V))$ is an abstract operator system.