A solution to Haagerup's problem and positive Hahn-Banach separation theorems in operator algebras

TL;DR

This paper resolves a long-standing question by Haagerup regarding the positive bipolar theorem in C*-algebras, leading to new positive Hahn-Banach separation theorems and simplifying previous solutions to Dixmier's problem.

Contribution

It provides the first positive bipolar theorem for dual spaces of C*-algebras and establishes four positive Hahn-Banach theorems across various operator algebra structures.

Findings

Confirmed the positive bipolar theorem for C*-algebra duals.

Established four positive Hahn-Banach separation theorems.

Simplified Haagerup's approach to Dixmier's problem.

Abstract

We affirmatively resolve a question posed by Uffe Haagerup in 1975 on the positive version of the bipolar theorem on the dual spaces of C$^*$-algebras. As a direct consequence, we obtain a complete set of four positive Hahn-Banach separation theorems on von Neumann algebras, their preduals, C$^*$-algebras, and their duals. Furthermore, with the idea used to solve the problem, we simplify Haagerup's original solution to Dixmier's problem on normal weights.