Crossed products by compact group actions with the weak tracial Rokhlin property
Xiaochun Fang, Haotian Tian
TL;DR
The paper develops a framework for compact group actions with a weak tracial Rokhlin property with comparison on simple unital C*-algebras and proves that key regularity properties—simplicity, pure infiniteness, and (tracial) $\mathcal{Z}$-stability, together with various comparison notions—pass from the original algebra to both the fixed point algebra and the crossed product via an averaging technique and Morita equivalence. Central to the approach is a central-sequence formulation and a technical theorem that constructs an invariant contraction in the fixed point algebra together with an equivariant order-zero map from $C(G)$ into the fixed-point orbit, enabling transfer of structure. The work also clarifies the relationship to finite-group generalizations, provides a nontrivial example on the Jiang-Su algebra with the group $G=(S_2)^{\mathbb{N}}$, and analyzes when the radius of comparison is preserved or bounded. Collectively, these results extend permanence properties under group actions beyond the Elliott program and expand the toolkit for studying crossed products by compact groups with Rokhlin-type properties.
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.