Compact Group Actions with the Tracial Rokhlin Property

TL;DR

This paper introduces a new 'tracial' Rokhlin property for compact group actions on C*-algebras, showing it preserves many structural properties and generalizes previous finite group cases, with diverse examples.

Contribution

It defines a tracial Rokhlin property for compact group actions on C*-algebras and demonstrates its preservation of key algebraic properties, extending prior finite group results.

Findings

Fixed point algebras retain simplicity and other properties.

The tracial Rokhlin property generalizes finite group cases.

Examples include actions on UHF, AT, and Cuntz algebras with this property.

Abstract

We define a "tracial" analog of the Rokhlin property for actions of second countable compact groups on infinite dimensional simple separable unital C*-algebras. We prove that fixed point algebras under such actions (and, in the appropriate cases, crossed products by such actions) preserve simplicity, Property (SP), tracial rank zero, tracial rank at most one, the Popa property, tracial Jiang-Su stability, Jiang-Su stability when the algebra is nuclear, infiniteness, and pure infiniteness. We also show that the radius of comparison of the fixed point algebra is no larger than that of the original algebra. Our version of the tracial Rokhlin property is an exact generalization of the tracial Rokhlin property for actions of finite groups on classifiable C*-algebras (in the sense of the Elliott program), but for actions of finite groups on more general C*-algebras it may be stronger. We discuss several alternative versions of the tracial Rokhlin property. We give examples of actions of a totally disconnected infinite compact group on a UHF algebra, and of the circle group on a simple unital AT algebra and on the Cuntz algebra ${\mathcal{O}}_{\infty}$, which have our version of the tracial Rokhlin property, but do not have the Rokhlin property, or even finite Rokhlin dimension with commuting towers.