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In this paper, let $A$ be a unital separable simple infinite dimensional C*-algebra. Let $\alpha\colon G\to \mathrm{Aut}(A)$ be an action of a second-countable compact group which has the tracial Rokhlin property with comparison. Then we prove that the fixed point algebra $A^\alpha$ has uniform property $\Gamma$ (and crossed product $A\rtimes_\alpha G$ has stablized property $\Gamma$). Let $\alpha\colon G\to \mathrm{Aut}(A)$ be an action of a finite group which has the weak tracial Rokhlin property. Then we prove that the crossed product $A\rtimes_\alpha G$ and fixed point algebra $A^\alpha$ have uniform property $\Gamma$.

In this paper, we give some properties of the fixed point algebra and the crossed product of a unital separable simple infinite dimensional C*-algebra by an action of a second-countable compact group with the tracial Rokhlin property with comparison that could be deduced from the properties of its original algebra: (1) stable rank one; (2) real rank zero; (3) $\beta$-comparison; (4) Winter's $n$-comparison; (5) $m$-almost divisible; (6) weakly ($m$,$n$)-divisible., Comment: final version, 24 pages. This version was accepted in Rocky Mountain Journal of Mathematics