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

Ikhan Choi

TL;DR

Haagerup's 1975 question on a positive bipolar theorem for dual spaces of C$^*$-algebras is resolved affirmatively. The authors develop a framework based on a one-parameter functional calculus and commutant Radon-Nikodym maps to derive four positive Hahn-Banach separation theorems across von Neumann algebras, preduals, C$^*$-algebras, and their duals. They also simplify Haagerup's approach to Dixmier's problem on normal weights and establish natural correspondences between weights and convex hereditary subsets. The results unify positive bipolar/separation phenomena in operator algebras and provide techniques that streamline classical questions about normal weights.

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.

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

TL;DR

Haagerup's 1975 question on a positive bipolar theorem for dual spaces of C-algebras is resolved affirmatively. The authors develop a framework based on a one-parameter functional calculus and commutant Radon-Nikodym maps to derive four positive Hahn-Banach separation theorems across von Neumann algebras, preduals, C-algebras, and their duals. They also simplify Haagerup's approach to Dixmier's problem on normal weights and establish natural correspondences between weights and convex hereditary subsets. The results unify positive bipolar/separation phenomena in operator algebras and provide techniques that streamline classical questions about normal weights.

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.

Paper Structure

This paper contains 5 sections, 8 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Suppression by the one-parameter family of functional calculi
  4. Bounded commutant Radon-Nikodym derivatives
  5. Proofs of theorems

Key Result

Theorem 1

Let $M$ be a von Neumann algebra, and let $A$ be a C$^*$-algebra.

Theorems & Definitions (16)

  • Theorem
  • Corollary
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 6 more