The radius of comparison of the crossed product by a tracially strictly approximately inner action

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 $\operatorname{Cu} \left(C^*(G, A, \alpha)\right)$. If $\alpha$ is strictly approximately inner, then in fact $\operatorname{Cu} (A) \to \operatorname{Cu} \left(C^*(G, A, \alpha) \right)$ 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) $\operatorname{rc} (A) =\operatorname{rc} \left(C^*(G, A, \alpha)\right)= \eta$.