Regularity of compact convex sets--classical and noncommutative

TL;DR

This paper extends the classical regularity theory of convex sets to complex and noncommutative settings, establishing new results on embeddings and duality in these advanced mathematical frameworks.

Contribution

It develops a comprehensive regularity theory for convex sets in complex locally convex spaces and noncommutative contexts, bridging classical and modern operator algebra approaches.

Findings

Classical regularity results are extended to complex LCTVS.

A new theory of regular embeddings for noncommutative convex sets is established.

Connections between classical convexity and noncommutative operator spaces are demonstrated.

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.