The radius of comparison of the crossed product by a weakly tracially strictly approximately inner action
Volume 271 / 2023
Abstract
Let $G$ be a finite group, let $A$ be an infinite-dimensional stably finite simple unital C$^*$-algebra, and let $\alpha \colon G \to {\rm Aut} (A)$ be a weakly tracially strictly approximately inner action of $G$ on $A$. Then the radius of comparison satisfies ${\rm rc} (A) \leq {\rm rc} (C^* (G, A, \alpha ) )$, and if $C^*(G, A, \alpha )$ is simple, then ${\rm rc} (A) \leq {\rm rc} ( C^* (G, A, \alpha ) ) \leq {\rm rc} (A^{\alpha })$. Further, the inclusion of $A$ in $C^*(G, A, \alpha )$ induces an isomorphism from the purely positive part of the Cuntz semigroup ${\rm Cu} (A)$ to its image in ${\rm Cu} (C^* (G, A, \alpha ))$. If $\alpha $ is strictly approximately inner, then in fact ${\rm Cu} (A) \to {\rm Cu} (C^* (G, A, \alpha ))$ is an ordered semigroup isomorphism onto its range. Also, for every finite group $G$ and for every $\eta \in (0, 1/{\rm card} (G))$, we construct a simple separable unital AH algebra $A$ with stable rank one and an approximately representable but pointwise outer action $\alpha \colon G \to {\rm Aut} (A)$ such that ${\rm rc} (A) ={\rm rc}(C^*(G, A, \alpha ))= \eta $.
