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Real Non-Commutative Convexity I

David P. Blecher, Caleb McClure

TL;DR

This work initiates and develops real noncommutative convexity by establishing foundational infrastructure for real nc convex sets and real operator systems, and by connecting them to the complex theory through a functorial complexification. It extends Kadison–Duff theory to the real nc setting via a real categorical duality, a real bipolar theorem, and Min/Max constructions, while carefully analyzing how complexification interacts with these structures. Key contributions include intrinsic/extrinsic complexification of real nc convex sets, a real version of the DK duality, and a real bipolar framework that parallels the complex theory. The results pave the way for real nc convex analysis, including real nc affine functions, real nc functions, and real operator-system envelopes, with applications to real matrix convexity and potential real Choquet-type theories in future work.

Abstract

We initiate the theory of real noncommutative (nc) convex sets, the real case of the recent and profound complex theory developed by Davidson and Kennedy. The present paper focuses on the real case of the topics from the first several sections of their memoir. Later results will be discussed in future papers. We develop here some of the infrastructure of real nc convexity, giving many foundational structural results for real operator systems and their associated nc convex sets, and elucidate how the complexification interacts with the basic convexity theory constructions. Several new features appear in the real case, including the novel notion of the complexification of a nc convex set.

Real Non-Commutative Convexity I

TL;DR

This work initiates and develops real noncommutative convexity by establishing foundational infrastructure for real nc convex sets and real operator systems, and by connecting them to the complex theory through a functorial complexification. It extends Kadison–Duff theory to the real nc setting via a real categorical duality, a real bipolar theorem, and Min/Max constructions, while carefully analyzing how complexification interacts with these structures. Key contributions include intrinsic/extrinsic complexification of real nc convex sets, a real version of the DK duality, and a real bipolar framework that parallels the complex theory. The results pave the way for real nc convex analysis, including real nc affine functions, real nc functions, and real operator-system envelopes, with applications to real matrix convexity and potential real Choquet-type theories in future work.

Abstract

We initiate the theory of real noncommutative (nc) convex sets, the real case of the recent and profound complex theory developed by Davidson and Kennedy. The present paper focuses on the real case of the topics from the first several sections of their memoir. Later results will be discussed in future papers. We develop here some of the infrastructure of real nc convexity, giving many foundational structural results for real operator systems and their associated nc convex sets, and elucidate how the complexification interacts with the basic convexity theory constructions. Several new features appear in the real case, including the novel notion of the complexification of a nc convex set.

Paper Structure

This paper contains 15 sections, 39 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operator Systems and Operator Spaces
  4. Noncommutative real convex sets and affine functions
  5. Some relations between the real and complex case
  6. Affine maps as an operator system
  7. Complexification
  8. Real Categorical Duality
  9. Bipolar Theorem
  10. Real Bipolar Theorem
  11. Max and Min nc convex sets
  12. Noncommutative Functions
  13. Real NC Functions
  14. Maximal C*-algebra
  15. Minimal $C^*$-algebra

Key Result

Proposition 2.4

Suppose we have $K$ a closed nc convex set over a dual operator space $E$, a net $\{x_i \in K_{n_i}\}$, and a net of isometries $\{\alpha_i \in \mathcal{M}_{n, n_i}\}$ such that $\lim \alpha_i \alpha_i^\mathsf{T} = 1_n$ and $\lim \alpha_i x_i \alpha_i^\mathsf{T} = x \in M_n(E)$. Then, $x \in K_n$.

Theorems & Definitions (77)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 67 more