Haagerup and St{\o}rmer's conjecture for pointwise inner automorphisms

TL;DR

This paper proves that type III_1 factors with trivial bicentralizer satisfy Haagerup and St{ }mer's conjecture, linking the conjecture to Connes' bicentralizer problem and using advanced operator algebra techniques.

Contribution

It establishes the conjecture for a broad class of factors, connecting it to the resolution of Connes' bicentralizer problem.

Findings

Type III_1 factors with trivial bicentralizer satisfy the conjecture.

The conjecture holds if Connes' bicentralizer problem is affirmatively resolved.

The proof employs Popa's intertwining theory and Marrakchi's work on relative bicentralizers.

Abstract

In 1988, Haagerup and St{\o}rmer conjectured that any pointwise inner automorphism of a type $\rm III_1$ factor is a composition of an inner and a modular automorphism. We study this conjecture and prove that any type $\rm III_1$ factor with trivial bicentralizer indeed satisfies this condition. In particular, this shows that Haagerup and St{\o}rmer's conjecture holds in full generality if Connes' bicentralizer problem has an affirmative answer. Our proof is based on Popa's intertwining theory and Marrakchi's recent work on relative bicentralizers.