Crossed products by compact group actions with the weak tracial Rokhlin property

TL;DR

This paper introduces a new class of compact group actions with the weak tracial Rokhlin property, generalizing previous concepts, and demonstrates how key algebraic properties are preserved under such actions, including an example involving the Jiang-Su algebra.

Contribution

It defines the weak tracial Rokhlin property for compact group actions and proves the transfer of several structural properties to crossed products, expanding the understanding of group actions in operator algebras.

Findings

Simplicity and pure infiniteness are preserved in crossed products.

The radius of comparison of fixed point algebras does not increase.

An example with $(S_2)^ N$ acting on the Jiang-Su algebra is provided.

Abstract

In this paper, we introduce compact group actions with the weak tracial Rokhlin property. This concept simultaneously generalizes finite group actions with the weak tracial Rokhlin property and compact group actions with the tracial Rokhlin property (in the sense of the Elliott program). Under this framework, we prove that simplicity, pure infiniteness, tracial $\mathcal{Z}$-stability and the combination of nuclearity and $\mathcal{Z}$-stability can be transferred from the original algebra to the crossed product. We also show that the radius of comparison of the fixed point algebra does not exceed that of the original algebra. Furthermore, we discuss the relationship between our definition and natural generalization of the finite group case in non-Elliott program settings. Finally, we provide a nontrivial example of a compact group action with the weak tracial Rokhlin property with comparison: an action of $(S_2)^\mathbb{N}$ on the Jiang-Su algebra $\mathcal{Z}$. Since $\mathcal{Z}$ contains no nontrivial projections, this action does not possess the tracial Rokhlin property.