The Dynamical Radius of Comparison for C*-Dynamical Systems

Abstract: We introduce a dynamical version $\operatorname{rc} (A, α)$ of the radius of comparison $\operatorname{rc} (A)$ of a unital C*-algebra, based on the dynamical Cuntz semigroup. We also give an intrinsic ordered semigroup definition, which agrees with $\operatorname{rc} (A, α)$ when $A$ is residually stably finite, and is lower semicontinuous for equivariant direct limits with injective unital maps. We construct actions $α$ of $G = \mathbb{Z} / 2 \mathbb{Z}$ on simple unital AH~algebras for which $\operatorname{rc} (A, α)$ lies strictly between $\operatorname{rc} (A)$ and $\operatorname{rc} (A) / \operatorname{card} (G)$, and actions $α$ of a finite group $G$ on unital C*-algebras for which $\operatorname{rc} (A, α)$ lies strictly between $\operatorname{rc} (A)$ and $\operatorname{rc} (C^* (G, A, α))$. For a minimal action of a countable discrete group $G$ on a zero dimensional compact metrizable space $X$, we prove that $\operatorname{rc} (C (X), α) = 0$ if and only if the action has dynamical comparison as defined by Kerr. For finite group actions on simple unital stably finite C*-algebras, assuming the weak tracial Rokhlin property, we get $$\operatorname{rc} (A, α) \leq \operatorname{rc} (A) / \operatorname{card} (G)$$, and assuming weak tracial strict approximate innerness, we get $\operatorname{rc} (A, α) = \operatorname{rc} (A)$.