Compact Group Actions with the Tracial Rokhlin Property II: Examples and Nonexistence Theorems

TL;DR

This paper explores examples and nonexistence results for actions of compact groups on C*-algebras with the tracial Rokhlin property, highlighting differences from classical Rokhlin properties and constructing specific cases.

Contribution

It provides new examples of group actions with the tracial Rokhlin property that lack the classical Rokhlin property, and establishes nonexistence results for certain actions.

Findings

Constructed examples of actions with the tracial Rokhlin property but not the Rokhlin property.

Identified actions of the circle group on simple unital AT algebras with the property.

Proved nonexistence of certain circle group actions on simple AF algebras with weak properties.

Abstract

In a previous paper, we introduced the restricted tracial Rokhlin property with comparison, a ``tracial'' analog of the Rokhlin property for actions of second countable compact groups on infinite dimensional simple separable unital C*-algebras. In this paper, we give three classes of examples of actions of compact groups which have this property but do not have the Rokhlin property, or even finite Rokhlin dimension with commuting towers. One class consists of infinite tensor products of finite group actions with the tracial Rokhlin property, giving actions of the product of the groups involved. The second class consists of actions of the circle group on simple unital AT~algebras. The construction of the third class starts with an action of the circle on the Cuntz algebra ${\mathcal{O}}_{\infty}$ which has the restricted tracial Rokhlin property with comparison; by contrast, it is known that there is no action of this group on ${\mathcal{O}}_{\infty}$ which has finite Rokhlin dimension with commuting towers. We can then tensor this action with the trivial action on any unital purely infinite simple separable nuclear C*-algebra. One also gets such actions on certain purely infinite simple separable nuclear C*-algebras by tensoring the AT~examples with the trivial action on ${\mathcal{O}}_{\infty}$; these are different. We also discuss other tracial Rokhlin properties for actions of compact groups, and prove that there is no direct limit action of the circle group on a simple AF~algebra which even has the weakest of these properties.