Cocycle perturbations and ergodicity for actions on type III factors

Yusuke Isono

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arXiv · 2512.12931

Cocycle perturbations and ergodicity for actions on type III factors

Yusuke Isono

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Abstract

We study cocycle perturbations of state preserving actions on type $\mathrm{III}_1$ factors. Extending the theorem of Marrakchi and Vaes for type $\mathrm{II}_1$ factors, we show that a state preserving outer $\mathbb Z$-action on a type $\mathrm{III}_1$ factor with trivial bicentralizer admits a unitary cocycle whose perturbation becomes an ergodic action. This partially answers a question of Marrakchi and Vaes. A major difference from the type $\mathrm{II}$ case is that the modular automorphism group naturally appears as part of the action, making the construction of the required cocycle more delicate.

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Yusuke Isono. 2025-12-15. Cocycle perturbations and ergodicity for actions on type III factors. https://arxiv.org/abs/2512.12931

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