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# The Jiang-Su absorption for inclusions of unital C*-algebras

Published:2017-12-06
Printed: Apr 2018
• Hiroyuki Osaka,
Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga, 525-8577 Japan
• Tamotsu Teruya,
Faculty of Education, Gunma University, 4-2 Aramaki-machi, Maebashi City, Gunma, 371-8510, Japan
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## Abstract

We introduce the tracial Rokhlin property for a conditional expectation for an inclusion of unital C*-algebras $P \subset A$ with index finite, and show that an action $\alpha$ from a finite group $G$ on a simple unital C*-algebra $A$ has the tracial Rokhlin property in the sense of N. C. Phillips if and only if the canonical conditional expectation $E\colon A \rightarrow A^G$ has the tracial Rokhlin property. Let $\mathcal{C}$ be a class of infinite dimensional stably finite separable unital C*-algebras which 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 $M_n(A) \in \mathcal{C}$. (3) If $A \in \mathcal{C}$ and $p \in A$ is a nonzero projection, then $pAp \in \mathcal{C}$. Suppose that any C*-algebra in $\mathcal{C}$ is weakly semiprojective. We prove that if $A$ is a local tracial $\mathcal{C}$-algebra in the sense of Fan and Fang and a conditional expectation $E\colon A \rightarrow P$ 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 $E\colon A \rightarrow P$ has the tracial Rokhlin property, then $P$ is Jiang-Su absorbing. As an application, when an action $\alpha$ from a finite group $G$ on a simple unital 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 $W(A)$ is hereditary to $W(P)$ if $A$ is simple, separable, exact, unital, and $E\colon A \rightarrow P$ has the tracial Rokhlin property.
 Keywords: Jiang-Su absorption, inclusion of C*-algebra, strict comparison
 MSC Classifications: 46L55 - Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20] 46L35 - Classifications of $C^*$-algebras

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