Rokhlin dimension for compact group actions

TL;DR

This paper introduces and studies Rokhlin dimension for compact group actions on C*-algebras, generalizing previous finite group notions and exploring implications for noncommutative freeness and K-theoretic obstructions.

Contribution

It extends the concept of Rokhlin dimension to compact groups, compares it with existing notions, and provides new K-theoretic obstructions, confirming a conjecture of Phillips.

Findings

Freeness for compact Lie group actions is equivalent to finite Rokhlin dimension.

Commuting towers cannot always be arranged, even without K-theoretic obstructions.

Confirmed Phillips' conjecture regarding K-theoretic obstructions.

Abstract

We introduce and systematically study the notion of Rokhlin dimension (with and without commuting towers) for compact group actions on $C^*$-algebras. This notion generalizes the one introduced by Hirshberg, Winter and Zacharias for finite groups, and contains the Rokhlin property as the zero dimensional case. We show, by means of an example, that commuting towers cannot always be arranged, even in the absence of $K$-theoretic obstructions. For a compact Lie group action on a compact Hausdorff space, freeness is equivalent to finite Rokhlin dimension of the induced action. We compare the notion of finite Rokhlin dimension to other existing definitions of noncommutative freeness for compact group actions. We obtain further $K$-theoretic obstructions to having an action of a non-finite compact Lie group with finite Rokhlin dimension with commuting towers, and use them to confirm a conjecture of Phillips.