Separation theorems for bounded convex sets of bounded operators

TL;DR

This paper introduces new metric characterizations for the closures of convex hulls of bounded sets in $C^*$-algebras and von Neumann algebras, with applications to majorization theory.

Contribution

It provides novel metric-based criteria for closures of convex sets in operator algebras, advancing the understanding of their structure.

Findings

New metric characterizations for norm and ultraweak closures

Applications to majorization theory in operator algebras

Enhanced understanding of convex hulls in $C^*$- and von Neumann$-algebras

Abstract

We establish new metric characterizations for the norm (respectively, ultraweak) closure of the convex hull of a bounded set in an arbitrary $C^*$-algebra (respectively, von Neumann algebra), and provide applications of these results to the majorization theory.