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 G Aut (A) be an action which has the weak tracial Rokhlin property. Let a be a non-zero positive element in A^ K. Then we show that the radius of comparison of Cu (A^) relative to a is bounded above by the radius of comparison of Cu (A) relative to a. If further A is exact and a is in the Pedersen ideal of A^ K, then the radius of comparison of Cu (A_ G) relative to a is equal to its radius of comparison relative to p a, scaled by 1/|G|, where p is the averaging projection in the multiplier algebra of (A K) ₈₃ G. Moreover, the radius of comparison of Cu (A_ G) relative to a is bounded above by 1/|G| times the radius of comparison of Cu (A) relative to a. We also prove that the inclusion of A^ in A induces an isomorphism from the purely positive part of the Cuntz semigroup Cu (A^) to the fixed point of the purely positive part of 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.