迹严格近似内作用交叉积的比较半径
The radius of comparison of the crossed product by a tracially strictly approximately inner action
AI总结：
本文研究有限群在单C*-代数上的迹严格近似内作用，证明了交叉积的比较半径介于原代数与其不动点代数之间，并构造了满足特定性质的例子。
摘要：
设$G$为有限群，$A$为无限维稳定有限单单位C*-代数，且$α\colon G \to \operatorname{Aut} (A)$为$G$在$A$上的迹严格近似内作用。则比较半径满足$\operatorname{rc} (A) \leq \operatorname{rc} \big( C^*(G, A, α) \big)$，且若$C^*(G, A, α)$是单的，则$\operatorname{rc} (A) \leq \operatorname{rc} \big( C^*(G, A, α) \big) \leq \operatorname{rc} (A^α)$。进一步，$A$在$C^*(G, A, α)$中的包含映射诱导了从Cuntz半群$\operatorname{Cu} (A)$的纯正部分到其在$\operatorname{Cu} \left(C^*(G, A, α)\right)$中像的同构。若$α$是严格近似内的，则实际上$\operatorname{Cu} (A) \to \operatorname{Cu} \left(C^*(G, A, α) \right)$是到其值域的序半群同构。此外，对每个有限群$G$及每个$η\in \left(0, \frac{1}{\operatorname{card} (G)}\right)$，我们构造一个稳定秩为一的简单可分单AH代数$A$及严格近似内作用$α\colon G \to \operatorname{Aut} (A)$，使得：(1) $α$是逐点外的且不具有弱迹Rokhlin性质。(2) $\operatorname{rc} (A) =\operatorname{rc} \left(C^*(G, A, α)\right)= η$。
英文摘要：
Let $G$ be a finite group, let $A$ be an infinite-dimensional stably finite simple unital C*-algebra, and let $α\colon G \to \operatorname{Aut} (A)$ be a tracially strictly approximately inner action of $G$ on $A$. Then the radius of comparison satisfies $\operatorname{rc} (A) \leq \operatorname{rc} \big( C^*(G, A, α) \big)$ and if $C^*(G, A, α)$ is simple, then $\operatorname{rc} (A) \leq \operatorname{rc} \big( C^*(G, A, α) \big) \leq \operatorname{rc} (A^α)$. Further, the inclusion of $A$ in $C^*(G, A, α)$ induces an isomorphism from the purely positive part of the Cuntz semigroup $\operatorname{Cu} (A)$ to its image in $\operatorname{Cu} \left(C^*(G, A, α)\right)$. If $α$ is strictly approximately inner, then in fact $\operatorname{Cu} (A) \to \operatorname{Cu} \left(C^*(G, A, α) \right)$ is an ordered semigroup isomorphism onto its range. Also, for every finite group $G$ and for every $η\in \left(0, \frac{1}{\operatorname{card} (G)}\right)$, we construct a simple separable unital AH algebra $A$ with stable rank one and a strictly approximately inner action $α\colon G \to \operatorname{Aut} (A)$ such that: (1) $α$ is pointwise outer and doesn't have the weak tracial Rokhlin property. (2) $\operatorname{rc} (A) =\operatorname{rc} \left(C^*(G, A, α)\right)= η$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
