The Rokhlin property for inclusions of $C^*$-algebras
The Rokhlin property for inclusions of $C^*$-algebras
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$C^*$-代数包含的 Rokhlin 性质
DOI：
10.1216/rmj.2020.50.1785
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发表时间：
2018
影响因子：
0.8
通讯作者：
T. Teruya
中科院分区：
文献类型：
--
作者：
H. Osaka;T. Teruya
设$P \subset A$是在Izumi意义上具有有限指标的$\sigma$ -单位C*-代数的一个包含。然后，我们为从$A$到$P$的条件期望$E$引入Rokhlin属性，并说明如果$A$很简单并且满足下面列出的任何属性$(1) \sim (12)$，并且$E$具有Rokhlin属性，那么$P$也具有Rokhlin属性。
(1)简洁性；(2)核能力；(3)吸收给定强自吸收C*代数$\mathcal{D}$的C*代数；(4)稳定秩为1的C*-代数；(5)实秩为0的C*-代数；(6)核维数最多为$n$的C*-代数，其中$n \in Z^+$；(7)分解的C*-代数最大为$n$，其中$n \in Z^+$；(8)与AF代数稳定同构的可分离简单C*-代数；(9)与AI代数稳定同构的可分离简单C*-代数；（10）与AT代数稳定同构的可分离简单C*-代数（11）一维NCCW配合物序直接极限稳定同构的可分离简单C*-代数；（12）具有正元严格比较的可分离C*代数。特别地，当$\alpha : G \rightarrow \rm{Aut}(A)$是具有Nawata意义上的Rokhlin性质的有限群$G$对$A$的作用时，属性$(1) \sim (12)$被继承到$A$的不动点代数$A^\alpha$和交叉积代数$A \rtimes_\alpha G$。
Let $P \subset A$ be an inclusion of $\sigma$-unital C*-algebras with a finite index in the sense of Izumi. Then we introduce the Rokhlin property for a conditional expectation $E$ from $A$ onto $P$ and show that if $A$ is simple and satisfies any of the property $(1) \sim (12)$ listed in the below, and $E$ has the Rokhlin property, then so does $P$.
(1) Simplicity;(2) Nuclearity;(3) C*-algebras that absorb a given strongly self-absorbing C*-algebra $\mathcal{D}$; (4)C*-algebras of stable rank one; (5) C*-algebras of real rank zero;(6) C*-algebras of nuclear dimension at most $n$, where $n \in Z^+$; (7)C*-algebras of decomposition rank at most $n$, where $n \in Z^+$; (8) Separable simple C*-algebras that are stably isomorphic to AF algebras; (9) Separable simple C*-algebras that are stably isomorphic to AI algebras; (10) Separable simple C*-algebras that are stably isomorphic to AT algebras; (11) Separable simple C*-algebras that are stably isomorphic to sequential direct limits of one dimensional NCCW complexes; (12) Separable C*-algebras with strict comparison of positive elements. In particular, when $\alpha : G \rightarrow \rm{Aut}(A)$ is an action of a finite group $G$ on $A$ with the Rokhlin property in the sense of Nawata, the properties $(1) \sim (12)$ are inherited to the fixed point algebra $A^\alpha$ and the crossed product algebra $A \rtimes_\alpha G$ from $A$.
影响因子：
2.5
作者：
Masaki Izumi
通讯作者：
Masaki Izumi