The Jiang–Su Absorption for Inclusions of Unital C*-algebras
The Jiang–Su Absorption for Inclusions of Unital C*-algebras
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DOI：
10.4153/cjm-2017-033-7
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发表时间：
2014-04
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影响因子：
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通讯作者：
H. Osaka;T. Teruya
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文献类型：
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作者：
H. Osaka;T. Teruya
本文引入了有限元单元$P、子集、A的包含条件期望的迹Rokhlin性质，并证明了有限群$G在单单位元${\Text{C}^{*}}$-代数上的作用$\α$具有N.C.Phillips意义下的迹Rokhlin性质当且仅当规范的条件期望$E：\，A\，\to，{{A}^{G}}，$拥有Trail Rokhlin财产。设$\数学{C}$是一类无穷维稳定有限可分的单位元${{\Text{C}}^{*}}$-代数，在下列条件下是闭的：(1)如果$A\，\in，\数学{C}$和$B\，\cong\，A$，则$B\，\in，\数学{C}$.(2)如果$A\，\In\，\Mathcal{C}$和$n\，\in\，\mathbb{N}$，则${{M}_{n}}\Left(A\Right)\，\In\，\Mathcal{C}$。(3)如果$A，\In，\Mathcal{C}$和$p，\in是非零投影，则$Pap\，\in，\Mathcal{C}$.设$\mathcal{C}$中的任意${{\Text{C}}^{*}}$-代数是弱半投射的。证明了：如果$A$是Fan和Fang意义下的局部迹${{\Text{C}}^{*}}$-代数，且有条件期望$E：\，A\，\to\，P$是具有迹Rokhlin性质的指数有限型，则$P$是单位局部迹$\数学{C}$-代数.主要结果是：如果$A$是简单的、可分的、单核的、酱-苏吸收的，且$E：、A、TO、P$具有迹Rokhlin性质，则$P$是姜-苏吸收的。作为应用，当有限群$G$在单单位元${\α{C}^{*}$-代数$A$上的作用具有迹罗克林性质时，则对$G$的任意子群$H$，不动点代数${{A}^{H}}$和交叉积代数$A{{\r次}_{{\α}_{|H}$$H$是蒋苏吸收的.我们还证明了如果$A$是简单的、可分的、精确的、单位的，且$E：\，A\，\to\，P$具有迹Rokhlin性质，则Cuntz半群$W\Left(A\Right)$的严格比较性质遗传于$W\Left(P\Right)$。
Abstract We introduce the tracial Rokhlin property for a conditional expectation for an inclusion of unital ${{\text{C}}^{*}}$ -algebras $P\,\subset \,A$ with index finite, and show that an action $\alpha$ from a finite group $G$ on a simple unital ${{\text{C}}^{*}}$ - algebra $A$ has the tracial Rokhlin property in the sense of N. C. Phillips if and only if the canonical conditional expectation $E:\,A\,\to \,{{A}^{G}}\,$ has the tracial Rokhlin property. Let $\mathcal{C}$ be a class of infinite dimensional stably finite separable unital ${{\text{C}}^{*}}$ -algebras that is closed under the following conditions: (1) If $A\,\in \,\mathcal{C}$ and $B\,\cong \,A$ , then $B\,\in \,\mathcal{C}$ . (2) If $A\,\in \,\mathcal{C}$ and $n\,\in \,\mathbb{N}$ , then ${{M}_{n}}\left( A \right)\,\in \,\mathcal{C}$ . (3) If $A\,\in \,\mathcal{C}$ and $p\,\in \,A$ is a nonzero projection, then $pAp\,\in \,\mathcal{C}$ . Suppose that any ${{\text{C}}^{*}}$ -algebra in $\mathcal{C}$ is weakly semiprojective. We prove that if $A$ is a local tracial ${{\text{C}}^{*}}$ -algebra in the sense of Fan and Fang and a conditional expectation $E:\,A\,\to \,P$ is of index-finite type with the tracial Rokhlin property, then $P$ is a unital local tracial $\mathcal{C}$ -algebra. The main result is that if $A$ is simple, separable, unital nuclear, Jiang–Su absorbing and $E:\,A\,\to \,P$ has the tracial Rokhlin property, then $P$ is Jiang–Su absorbing. As an application, when an action α from a finite group $G$ on a simple unital ${{\text{C}}^{*}}$ -algebra $A$ has the tracial Rokhlin property, then for any subgroup $H$ of $G$ the fixed point algebra ${{A}^{H}}$ and the crossed product algebra $A{{\rtimes }_{{{\alpha }_{|H}}}}$ $H$ is Jiang–Su absorbing. We also show that the strict comparison property for a Cuntz semigroup $W\left( A \right)$ is hereditary to $W\left( P \right)$ if $A$ is simple, separable, exact, unital, and $E:\,A\,\to \,P$ has the tracial Rokhlin property.