Cocycle Conjugacy of Free Bogoljubov Actions of $\mathbb{R}$

TL;DR

This paper proves that certain free Bogoljubov actions of the real line on the free group factor are cocycle conjugate precisely when their associated representations are conjugate, establishing a classification criterion.

Contribution

It establishes a complete classification of specific Bogoljubov actions on free group factors via conjugacy of their underlying representations.

Findings

Cocycle conjugacy of Bogoljubov actions corresponds to conjugacy of representations.

The result applies to sums of infinite multiplicity trivial and mixing representations.

Provides a classification framework for these actions on free group factors.

Abstract

We show that Bogoljubov actions of $\mathbb{R}$ on the free group factor $L(\mathbb{F}_{\infty})$ associated to sums of infinite multiplicity trivial and certain mixing representations are cocycle conjugate if and only if the underlying representations are conjugate.