Partial C*-dynamics and Rokhlin dimension

TL;DR

This paper extends the concept of Rokhlin dimension to partial actions of finite groups on C*-algebras, revealing new phenomena and preserving key structural properties in the partial setting, with implications for globalizable actions and topological dynamics.

Contribution

It introduces Rokhlin dimension for partial actions, analyzes its properties, and compares it with the global case, providing new insights and results in C*-algebra dynamics.

Findings

Structural properties like finite nuclear dimension are preserved under crossed products.

Finiteness of Rokhlin dimension is equivalent to freeness for topological partial actions.

Some results are new even for global group actions.

Abstract

We develop the notion of Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming from the fact that virtually all averaging arguments for finite group actions completely break down for partial systems. For example, fixed point algebra and crossed product are not in general Morita equivalent, and there is in general no local approximation of the crossed product $A\rtimes G$ by matrices over $A$. By using decomposition arguments for partial actions of finite groups, we show that a number of structural properties are preserved by formation of crossed products, including finite stable rank, finite nuclear dimension, and absorption of a strongly self-absorbing $C^*$-algebra. For partial actions with the Rokhlin property, being an AF-algebra is also preserved. Some of our results are new even in the global case. We also study the Rokhlin dimension of globalizable actions: while in general it differs from the Rokhlin dimension of its globalization, we show that they agree if the coefficient algebra is unital. For topological partial actions on spaces of finite covering dimension, we show that finiteness of the Rokhlin dimension is equivalent to freeness, thus providing a large class of examples to which our theory applies.