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Regularity of compact convex sets--classical and noncommutative

David P. Blecher

Abstract

The classical theory of regularity of embeddings of compact convex sets was developed in the 1970s, exclusively in the real case, and even there it does not appear to have been stated in its simplest form. We begin by revisiting this setting, showing that under a reasonable condition, every locally convex topological vector space $E$ that contains and is spanned by a compact convex set lying in a hyperplane not passing through the origin, is a (specific) dual Banach space equipped with the weak* topology. Second, we establish the corresponding regularity theory for convex sets in complex LCTVS's. Third, we develop a theory of regular embeddings for complex noncommutative convex sets, in the sense of Davidson and Kennedy. Finally, we use the complex theory to derive a theory of regular embeddings for real noncommutative convex sets. Interestingly, at present there appears to be no direct route to the latter.

Regularity of compact convex sets--classical and noncommutative

Abstract

The classical theory of regularity of embeddings of compact convex sets was developed in the 1970s, exclusively in the real case, and even there it does not appear to have been stated in its simplest form. We begin by revisiting this setting, showing that under a reasonable condition, every locally convex topological vector space that contains and is spanned by a compact convex set lying in a hyperplane not passing through the origin, is a (specific) dual Banach space equipped with the weak* topology. Second, we establish the corresponding regularity theory for convex sets in complex LCTVS's. Third, we develop a theory of regular embeddings for complex noncommutative convex sets, in the sense of Davidson and Kennedy. Finally, we use the complex theory to derive a theory of regular embeddings for real noncommutative convex sets. Interestingly, at present there appears to be no direct route to the latter.
Paper Structure (8 sections, 21 theorems, 29 equations)

This paper contains 8 sections, 21 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Regularity
  3. The real classical case
  4. Complex classical regularity
  5. Nc regularity
  6. Nc spans
  7. Complex nc regularity
  8. Real nc regularity

Key Result

Theorem 1.1

Suppose that $K$ is a compact matrix convex set in the sense of WW, in a complex LCTVS $E$. The embedding $\delta : K_1 \to \mathbb{A}(K)^*$ extends to a topological isomorphism $E \cong \mathbb{A}(K)^*$ which is a homeomorphism at all finite matrix levels if and only if $E = {\rm Span}(K_1)$ and ev

Theorems & Definitions (39)

  • Theorem 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • ...and 29 more