The relative radius of comparison of the crossed product of a non-unital C*-algebra by a finite group
M. Ali Asadi-Vasfi, George A. Elliott
TL;DR
This work extends the radius-of-comparison framework to non-unital C*-algebras under finite-group actions with the weak tracial Rokhlin property. It proves that the relative radius of comparison for the fixed-point algebra does not exceed that of the original algebra, and, under exactness and Pedersen-ideal hypotheses, yields a precise $1/|G|$-scaling formula for the relative radius of the crossed product. A key advance is the isomorphism on purely positive elements induced by the fixed-point inclusion ${\operatorname{Cu}}(A^{\alpha}) \to {\operatorname{Cu}}(A)^{\alpha}$, which standardizes comparison and traces in the non-unital setting. The results provide new structural insights into the Cuntz semigroups of crossed products and fixed-point algebras, with concrete applications to non-unital stably finite and non-type I algebras.
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 $α\colon G \to {\operatorname{Aut}} (A)$ be an action which has the weak tracial Rokhlin property. Let $a$ be a non-zero positive element in $A^α\otimes \mathcal{K}$. Then we show that the radius of comparison of $\operatorname{Cu} (A^α)$ 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 $A^α\otimes \mathcal{K}$, then the radius of comparison of $\operatorname{Cu} (A\rtimes_α G)$ 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 $(A \otimes \mathcal{K}) \rtimes_{α\otimes \operatorname{id}} G$. Moreover, the radius of comparison of $\operatorname{Cu} (A\rtimes_α G)$ 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 $A^α$ in $A$ induces an isomorphism from the purely positive part of the Cuntz semigroup ${\operatorname{Cu}} (A^α)$ 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.