Extension of the Marrakchi–Vaes ergodic cocycle results to type III factors (state-preserving case)

Establish whether the Marrakchi–Vaes results on ergodic cocycle perturbations for outer actions of countable amenable groups on type II1 factors extend to outer actions on type III factors, in particular within the state‑preserving setting where the action leaves a faithful normal state invariant.

Background

Marrakchi and Vaes proved in the type II1 setting that every outer action of a countable amenable group admits an ergodic cocycle and that such cocycles are generic. They posed an explicit open problem asking whether analogous statements hold for actions on type III factors.

This paper provides affirmative results for type III1 factors under a state‑preserving assumption and a trivial bicentralizer condition, indicating that the state‑preserving case is tractable and partially resolving the cited open problem.