On compact extensions of tracial $W^*$-dynamical systems
Abstract
We establish several classification results for compact extensions of tracial -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 -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.
Cite
@article{arxiv.2209.10238,
title = {On compact extensions of tracial $W^*$-dynamical systems},
author = {Asgar Jamneshan and Pieter Spaas},
journal= {arXiv preprint arXiv:2209.10238},
year = {2025}
}
Comments
[v4]: After publication, the authors found three issues in the original paper; we refer to the erratum, to appear in Groups Geom. Dyn. The main results remain unchanged, and this version is the entire paper containing all the corrections outlined in the erratum. [v3]: Published version. [v2]: after feedback updated references (see addendum in introduction) and corrected a mistake (see Remark 4.2)