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abstract: |
    Let $G$ be a finite group, let $A$ be an infinite-dimensional stably finite simple unital C\*-algebra, and let $\alpha \colon G \to {{\operatorname{Aut}}}(A)$ be a tracially strictly approximately inner action of $G$ on $A$. Then the radius of comparison satisfies ${{\operatorname{rc}}}(A) \leq   {{\operatorname{rc}}}\big( {C^* (G, A, {\alpha})}\big)$ and if $C^*(G, A, \alpha)$ is simple, then ${{\operatorname{rc}}}(A) \leq   {{\operatorname{rc}}}\big( {C^* (G, A, {\alpha})}\big) \leq {{\operatorname{rc}}}(A^{\alpha})$. Further, the inclusion of $A$ in $C^*(G, A, \alpha)$ induces an isomorphism from the purely positive part of the Cuntz semigroup ${{\operatorname{Cu}}}(A)$ to its image in ${{\operatorname{Cu}}}({C^* (G, A, {\alpha})})$. If $\alpha$ is strictly approximately inner, then in fact ${{\operatorname{Cu}}}(A) \to {{\operatorname{Cu}}}({C^* (G, A, {\alpha})})$ is an ordered semigroup isomorphism onto its range. Also, for every finite group $G$ and for every $\eta \in \left(0, \frac{1}{{{\operatorname{card}}}(G)}\right)$, we construct a simple separable unital AH algebra $A$ with stable rank one and a strictly approximately inner action $\alpha \colon G \to {{\operatorname{Aut}}}(A)$ such that:

    1.  $\alpha$ is pointwise outer and doesn’t have the weak tracial Rokhlin property.

    2.  ${{\operatorname{rc}}}(A) ={{\operatorname{rc}}}\left(C^*(G, A, \alpha)\right)= \eta$.

author:
- 'M. Ali Asadi-Vasfi'
title: ' The radius of comparison of the crossed product by a tracially strictly approximately inner action.'
