Compact convex sets and bases--classical and noncommutative

Abstract

Matrix and noncommutative convexity constitute an important area of modern noncommutative analysis and have found significant applications in mathematical physics. In the first part of our paper we give an abstract characterization of matrix convex sets, and compact matrix convex sets. Our approach is in some part via a universal Banach space (resp.\ operator space) $X_K$ of an abstract compact convex set (resp.\ matrix convex set) $K$. This turns out to be a concrete construction of the base norm space (resp.\ nc base norm space) with base $K$, together with a natural TVS topology. Noncommutative (nc for short) base norm spaces, recently developed by the first author and Hay, are an important class of operator spaces which include duals and preduals of unital $C^*$-algebras and von Neumann algebras, and operator systems, where the `base' is exactly the noncommutative convex set of (matrix) states on these. In the later parts of the paper we give many applications, mostly to base norm spaces (classical and noncommutative). We also refine some of our recent results concerning regularity of convex sets (classical and noncommutative). We give several interesting characterizations of base norm spaces (classical and noncommutative). Any such characterization will correspond by duality to a new characterization of operator systems, or in the classical case, of function systems. For example, (complex) nc dual base norm spaces are the matrix ordered LCTVS's $V$ such that $V$ (at level 1) has a linear base which is compact.