The radius of comparison of the crossed product by a tracially strictly approximately inner action

Abstract

Let $G$ be a finite group, let $A$ be an infinite-dimensional stably finite simple unital C*-algebra, and let $\alpha :G\to \mathrm{Aut}(A)$ be a tracially strictly approximately inner action of $G$ on $A$. Then the radius of comparison satisfies $\mathrm{rc}(A)\le \mathrm{rc}(C^{\ast }(G,A,\alpha ))$ and if $C^{\ast }(G,A,\alpha )$ is simple, then $\mathrm{rc}(A)\le \mathrm{rc}(C^{\ast }(G,A,\alpha ))\le \mathrm{rc}(A^{\alpha })$. Further, the inclusion of $A$ in $C^{\ast }(G,A,\alpha )$ induces an isomorphism from the purely positive part of the Cuntz semigroup $\mathrm{Cu}(A)$ to its image in $\mathrm{Cu}\left(C^{\ast }(G,A,\alpha )\right)$. If $\alpha$ is strictly approximately inner, then in fact $\mathrm{Cu}(A)\to \mathrm{Cu}\left(C^{\ast }(G,A,\alpha )\right)$ is an ordered semigroup isomorphism onto its range. Also, for every finite group $G$ and for every $\eta \in \left(0,\frac{1}{\mathrm{card}(G)}\right)$, we construct a simple separable unital AH algebra $A$ with stable rank one and a strictly approximately inner action $\alpha :G\to \mathrm{Aut}(A)$ such that: (1) $\alpha$ is pointwise outer and doesn't have the weak tracial Rokhlin property. (2) $\mathrm{rc}(A)=\mathrm{rc}\left(C^{\ast }(G,A,\alpha )\right)=\eta$.

35 Pages

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