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Real Noncommutative Convexity II: Extremality and nc convex functions

David P. Blecher, Caleb Becker McClure

Abstract

We continue the development of real noncommutative (nc) convexity, building on the recent and profound complex theory of Davidson and Kennedy. The present 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 a nc convex function and of the complexification of the convex envelope of a nc function. With an appendix by T. Russell.

Real Noncommutative Convexity II: Extremality and nc convex functions

Abstract

We continue the development of real noncommutative (nc) convexity, building on the recent and profound complex theory of Davidson and Kennedy. The present 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 a nc convex function and of the complexification of the convex envelope of a nc function. With an appendix by T. Russell.

Paper Structure

This paper contains 6 sections, 41 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Extreme, pure, maximal points, and the nc Choquet boundary
  3. Complexification of extremals
  4. Noncommutative convex functions
  5. Convex multivalued functions and convex envelopes
  6. Appendix by Travis Russell: A more detailed proof of Theorem \ref{['DK643']}.

Key Result

Theorem 2.4

A ucp map $u : V \to B(H)$ is maximal if and only if it has the unique extension property (UEP). Namely for every (or for some) $C^*$-algebra $B$ generated by a unital completely order isomorphic copy of $V$, there is a unique ucp extension $u : B \to B(H)$, and this extension is a $*$-homomorphism.

Theorems & Definitions (75)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • Corollary 2.7
  • proof
  • ...and 65 more