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arXiv · 2309.05279

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

Yusuke Isono
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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.

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Yusuke Isono. 2023-09-11. Haagerup and St{\o}rmer's conjecture for pointwise inner automorphisms. https://arxiv.org/abs/2309.05279

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