Cocycle perturbations and ergodicity for actions on type III factors

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.

19 pages