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

Abstract

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 $\alpha :G\to \mathrm{Aut}(A)$ 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 $\mathrm{Cu}(A^{\alpha })$ relative to $[a]$ is bounded above by the radius of comparison of $\mathrm{Cu}(A)$ 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 $\mathrm{Cu}(A{⋊}_{\alpha }G)$ relative to $[a]$ is equal to its radius of comparison relative to $[p\cdot a]$, scaled by $\frac{1}{|G|}$, where $p$ is the averaging projection in the multiplier algebra of $(A\otimes \mathcal{K}){⋊}_{\alpha \otimes \mathrm{id}}G$. Moreover, the radius of comparison of $\mathrm{Cu}(A{⋊}_{\alpha }G)$ relative to $[a]$ is bounded above by $\frac{1}{|G|}$ times the radius of comparison of $\mathrm{Cu}(A)$ 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 $\mathrm{Cu}(A^{\alpha })$ to the fixed point of the purely positive part of $\mathrm{Cu}(A)$. 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.