Cocycle perturbations and ergodicity for actions on type III factors

TL;DR

This paper extends cocycle perturbation techniques to type III_1 factors, showing that certain outer actions can be perturbed to become ergodic, with implications for understanding automorphism groups in operator algebras.

Contribution

It generalizes results from type II_1 to type III_1 factors, incorporating modular automorphisms into cocycle perturbation methods.

Findings

Existence of unitary cocycles making actions ergodic

Extension of Marrakchi and Vaes theorem to type III_1

Partial answer to Marrakchi and Vaes' question

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.