The relative radius of comparison of the crossed product of a non-unital C*-algebra by a finite group

In this paper, we prove results on the relative radius of comparison of C*-algebras and their crossed products, focusing on the non-unital setting. More precisely, let $A$ be a stably finite simple non-type-I (not necessarily unital) C*-algebra, let $G$ be a finite group, and let $\alpha \colon G \to {\operatorname{Aut}} (A)$ be an action which has the weak tracial Rokhlin property. Let $a$ be a non-zero positive element in $A^\alpha \otimes \mathcal{K}$. Then we show that the radius of comparison of $\operatorname{Cu} (A^\alpha)$ relative to $[a]$ is bounded above by the radius of comparison of $\operatorname{Cu} (A)$ relative to $[a]$. If further $A$ is exact and $a$ is in the Pedersen ideal of $A^\alpha \otimes \mathcal{K}$, then the radius of comparison of $\operatorname{Cu} (A\rtimes_{\alpha} G)$ relative to $[a]$ is equal to its radius of comparison relative to $[p\cdot a]$, scaled by $1/|G|$, where $p$ is the averaging projection in the multiplier algebra of $(A \otimes \mathcal{K}) \rtimes_{\alpha \otimes \operatorname{id}} G$. Moreover, the radius of comparison of $\operatorname{Cu} (A\rtimes_{\alpha} G)$ relative to $[a]$ is bounded above by $1/|G|$ times the radius of comparison of $\operatorname{Cu} (A)$ relative to $[a]$. We also prove that the inclusion of $A^{\alpha}$ in $A$ induces an isomorphism from the purely positive part of the Cuntz semigroup ${\operatorname{Cu}} (A^{\alpha})$ to the fixed point of the purely positive part of ${\operatorname{Cu}} (A)$. An important consequence of our results is that they apply to non-unital C*-algebras and give new insights into comparison theory of C*-algebras and their crossed products.