Journal of Operator Theory

Volume 96, Issue 1, Summer 2026 pp. 185-213.

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

Authors: Xiaochun Fang (1), Zhongli Wang (2)
Author institution: (1) School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University, Shanghai, China
(2) School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University, Shanghai, China

Summary: Let $A$ be a simple non-elementary $C^*$-algebra, and let $\alpha\colon G\rightarrow \mathrm{Aut}(A)$ be an action of a finite group $G$ on $A$ which has the weak tracial Rokhlin property. We prove the following results: (1) If $A$ is unital and has property (TM) introduced by Fu and Lin, then $A\rtimes_\alpha G$ has property (TM). As a corollary, if $A$ is unital, separable, and has stable rank one and strict comparison of positive elements, then $A\rtimes_\alpha G$ and $A^\alpha$ have stable rank one.\break (2) If $A$ has $m$-comparison, then $A\rtimes_\alpha G$ and $A^\alpha$ have $m$-comparison. (3) If $A$ is $n$-almost divisible, then $A\rtimes_\alpha G$ and $A^\alpha$ are $(2n+1)$-almost divisible.

DOI: http://dx.doi.org/10.7900/jot.2025mar13.2504
Keywords: $C^*$-algebra, crossed product, stable rank one, comparison, divisibility

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