Ergodic states on type III$_1$ factors and ergodic actions

Amine Marrakchi; Stefaan Vaes

doi:10.1515/crelle-2024-0007

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arXiv · 2305.14217

Ergodic states on type III$_1$ factors and ergodic actions

Amine Marrakchi·Stefaan Vaes

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Abstract

Since the early days of Tomita-Takesaki theory, it is known that a von Neumann algebra $M$ that admits a state $\varphi$ with trivial centralizer $M_\varphi$ must be a type III$_1$ factor, but the converse remained open. We solve this problem and prove that such ergodic states form a dense $G_\delta$ set among all faithful normal states on any III$_1$ factor with separable predual. Through Connes' Radon-Nikodym cocycle theorem, this problem is related to the existence of ergodic cocycle perturbations for outer group actions, which we consider in the second part of the paper.

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BibTeX RIS

Amine Marrakchi, Stefaan Vaes. 2023-05-23. Ergodic states on type III$_1$ factors and ergodic actions. https://doi.org/10.1515/crelle-2024-0007

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