Rokhlin Dimension for Compact Group Actions

We introduce and study the notion of Rokhlin dimension (with and without commuting towers) for compact group actions on sigma-unital C*-algebras. This notion generalizes the one introduced by Hirshberg, Winter, and Zacharias for finite groups and unital C*-algebras, 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. Furthermore, we obtain a Rokhlin-dimensional inequality that allows us to show that every pointwise outer action of a finite group on a Kirchberg algebra has Rokhlin dimension at most one.