Compact Group Actions with the Tracial Rokhlin Property II: Examples and Nonexistence Theorems
Javad Mohammadkarimi, N. Christopher Phillips
TL;DR
The paper develops and analyzes variants of Rokhlin-type properties for actions of compact groups on simple unital C*-algebras, focusing on the restricted tracial Rokhlin property with comparison and its strong/modified variants. It constructs concrete examples where this property holds without finite Rokhlin dimension with commuting towers, including infinite tensor products of finite-group actions, an $S^1$-action on a simple AT algebra, and an $S^1$-action on ${ m O}_{ ext{∞}}$ (and related tensorings on Kirchberg algebras). It also proves nonexistence results for certain direct limit actions (e.g., $S^1$ on AF algebras) and analyzes how these properties interact with crossed products and fixed-point algebras via approximate equivariant central maps and equivariant K-theory. The work highlights the broader prevalence of restricted tracial Rokhlin-type properties, clarifies the landscape between different Rokhlin notions, and raises several guiding open problems for future research in the structure and classification of group actions on C*-algebras.
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.