Nonunital operator systems and noncommutative convexity

TL;DR

This paper establishes a duality between nonunital operator systems and noncommutative convex sets, extending previous results and applying this to characterize C*-algebras, develop quotient theory, and relate to group properties.

Contribution

It introduces a duality framework for nonunital operator systems and applies it to new characterizations and extensions in noncommutative convexity and operator algebra theory.

Findings

Duality between operator systems and nc convex sets established.

Characterization of C*-algebras via nc quasistate spaces.

Development of quotient theory for operator systems.

Abstract

We establish the dual equivalence of the category of (potentially nonunital) operator systems and the category of pointed compact nc (noncommutative) convex sets, extending a result of Davidson and the first author. We then apply this dual equivalence to establish a number of results about operator systems, some of which are new even in the unital setting. For example, we show that the maximal and minimal C*-covers of an operator system can be realized in terms of the C*-algebra of continuous nc functions on its nc quasistate space, clarifying recent results of Connes and van Suijlekom. We also characterize "C*-simple" operator systems, i.e. operator systems with simple minimal C*-cover, in terms of their nc quasistate spaces. We develop a theory of quotients of operator systems that extends the theory of quotients of unital operator algebras. In addition, we extend results of the first author and Shamovich relating to nc Choquet simplices. We show that an operator system is a C*-algebra if and only if its nc quasistate space is an nc Bauer simplex with zero as an extreme point, and we show that a second countable locally compact group has Kazhdan's property (T) if and only if for every action of the group on a C*-algebra, the set of invariant quasistates is the quasistate space of a C*-algebra.