Nonunital operator systems and noncommutative convexity

Abstract

We establish the dual equivalence of the category of (potentially non-unital) 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. ContentsM. KENNEDY, S.J. KIM, AND N. MANOR 5. Pointed noncommutative functions 23 6. Minimal and maximal C*-covers 24 7. Characterization of unital operator systems 29 8. Quotients of operator systems 31 9. Noncommutative faces 35 10. C*-simplicity 37 11. Characterization of C*-algebras 39 12. Stable equivalence 42 13. Dynamics and Kazhdan's property (T) 43 References 47

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)]…”

Morita equivalence for operator systems

“…For an operator system S, there exists a unital completely isometric map ι S : S → B(H) so that the (so called maximal) C*-algebra C * max (S) := C * (ι S (S)) has the following (universal) property: for every unital completely positive map φ : S → B(K) there exists a * -homomorphism φ : C * (ι S (S)) → C * (φ(S)) such that φ•ι S = φ. The latter object was introduced by E. Kirchberg and S. Wasserman [47], and was recently studied in the larger category of "non-unital operator systems" (that is, matricially ordered operator spaces that admit a completely isometric complete order embedding into B(H) for some Hilbert space H) in [46].…”

“…Arveson in [1] -have played a cornerstone role in building quantised functional analysis [25,49], and are currently enjoying a surge of interest, both from a purely theoretical perspective [21,44,45,46] and in applications to the area of quantum information theory, where they are studied as non-commutative graphs [24,57,14,58,15]. Stable isomorphism of operator systems has further proved essential in recent developments in non-commutative geometry and mathematical physics [20].…”

Facial structure of matrix convex sets

“…(e) Note that a matrix multiface F of either type must contain all the matrix extreme points, whose proper matrix convex combinations describe the elements of F . In particular, the matrix interval [aI, bI] := ([aI n , bI n ]) n∈N has very few matrix convex multifaces, i.e., ∅, mconv({a}), mconv({b}) and [aI, bI] (see [KKM+,Example 9.9] for the fact that mconv({a}) and mconv({b}) are indeed matrix convex multifaces), and accordingly, its non-matrix convex matrix multifaces are of the form F in the notation from part (a), where F is a matrix face.…”