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 {\frac{1}{\lvert G\rvert}} , 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 {\frac{1}{\lvert G \rvert}} 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 for C*-algebras and their crossed products.