On compact extensions of tracial $W^*$-dynamical systems

TL;DR

This paper develops classification results for compact extensions of tracial W*-dynamical systems, addressing open questions and establishing non-commutative analogues of classical ergodic theory structures.

Contribution

It provides new classification theorems for compact extensions and joinings, and constructs non-commutative analogues of the Host-Kra-Ziegler tower.

Findings

Classification results for compact extensions of W*-dynamical systems

Resolution of open questions by Austin, Eisner, Tao, Duvenhage, and King

Existence of a non-commutative Host-Kra-Ziegler tower

Abstract

We establish several classification results for compact extensions of tracial $W^*$-dynamical systems and for relatively independent joinings thereof for actions of arbitrary discrete groups. We use these results to answer a question of Austin, Eisner, and Tao and some questions raised by Duvenhage and King. Moreover, combining our results with an earlier classification of weakly mixing extensions by Popa, we can derive non-commutative Furstenberg-Zimmer type dichotomies on the $L^2$-level. Although in general an adequate generalization of the Furstenberg-Zimmer tower of intermediate compact extensions doesn't seem possible in the von Neumann algebraic framework, we show that there always exists a non-commutative analogue of the finer Host-Kra-Ziegler tower for any ergodic action of a countable abelian group.