Real Noncommutative Convexity. II: Extremality and Noncommutative Convex Functions

We continue the development of real noncommutative (nc) convexity, building on the recent and profound complex theory of Davidson and Kennedy. This paper focuses on the theory of nc extreme points (and pure and maximal points) and the nc Choquet boundary in the real setting, as well as on the theory of real nc convex and semicontinuous functions and real nc convex envelopes. Our main emphasis is on how these notions interact with complexification. In particular, parts of the paper analyze in detail how various notions of “extreme” or “maximal” relate to our earlier concept of the complexification of a convex set. Several new features emerge in the real case, especially in the later sections, including the novel notions of the complexification of an nc convex function and of the complexification of the convex envelope of an nc function. With an Appendix by Russell.