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

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 ${\mathcal{O}}_{\infty}$