The Jiang–Su Absorption for Inclusions of Unital C*-algebras

Osaka, Hiroyuki ; Teruya, Tamotsu

Canadian journal of mathematics, Tome 70 (2018) no. 2, pp. 400-425

Résumé

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 followingconditions:(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.

Détail

DOI : 10.4153/CJM-2017-033-7

Mots-clés : 46L55, 46L35, Jiang–Su absorption, inclusion of C*-algebra, strict comparison