Quantization of $A_0(K)$-Spaces

Abstract

In this paper, we study $L^1$-matrix convex sets $\{K_n\}$ in $\ast$-locally convex spaces and show that every C${}^{\ast }$-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.

15 pages