Nonunital Operator Systems and Noncommutative Convexity

Abstract

We establish the dual equivalence of the category of generalized (i.e., potentially nonunital) operator systems and the category of pointed compact noncommutative (nc) convex sets, extending a result of Davidson and the 1st author. We then apply this dual equivalence to establish a number of results about generalized operator systems, some of which are new even in the unital setting. For example, we show that the maximal and minimal C*-covers of a generalized operator system can be realized in terms of theC*-algebra of continuous nc functions on its nc quasistate space, clarifying recent results of Connes and van Suijlekom. We also characterize “C*-simple” generalized operator systems, that is, generalized operator systems with a simple minimal C*-cover, in terms of their nc quasistate spaces. We develop a theory of quotients of generalized operator systems that extends the theory of quotients of unital operator systems. In addition, we extend results of the 1st author and Shamovich relating to nc Choquet simplices. We show that a generalized 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.

Tolerance relations and operator systems

“…The main conclusion from [33] is then that E ♯ is a unital operator system and that E is a non-unital operator system iff the embedding ı : E → E ♯ is a complete isometry. For a convex geometric description of operator systems and their unitization we refer to [7,19]. The following result is a special case of [33,Proposition 4.16(a)] Lemma 2.4.…”