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Strongly self-absorbing property for inclusions of $ \mathrm{C}^*$-algebras with a finite Watatani index


Authors: Hiroyuki Osaka and Tamotsu Teruya
Journal: Trans. Amer. Math. Soc. 366 (2014), 1685-1702
MSC (2010): Primary 46L55; Secondary 46L35
DOI: https://doi.org/10.1090/S0002-9947-2013-05907-7
Published electronically: September 13, 2013
MathSciNet review: 3145747
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Abstract: Let $ P \subset A$ be an inclusion of unital $ \mathrm {C}^*$-algebras and $ E\colon A \rightarrow P$ be a conditional expectation of index-finite type. We introduce a Rokhlin property for $ E$ and discuss the $ \mathcal {D}$-absorbing property, where $ \mathcal {D}$ is a separable, unital, strongly self-absorbing $ \mathrm {C}^*$-algebra, i.e., $ \mathcal {D} \not = \mathbb{C}$, and there exists an isomorphism $ \varphi \colon \mathcal {D} \rightarrow \mathcal {D} \otimes \mathcal {D}$ such that $ \varphi $ is approximately unitarily equivalent to the embedding $ d \mapsto d \otimes 1_D$. UHF algebras of infinite type, the Jiang-Su algebra $ \mathcal {Z}$, and Cuntz algebras $ O_2$ and $ O_\infty $ are typical examples of strongly self-absorbing $ \mathrm {C}^*$-algebras. In this paper, we consider permanence properties for a strongly self-absorbing property under inclusions of unital $ \mathrm {C}^*$-algebras with a finite Watatani index. We show the following:

Let $ P \subset A$ be an inclusion of unital $ \mathrm {C}^*$-algebras, $ E$ a conditional expectation from $ A$ onto $ P$ with a finite index, and $ \mathcal {D}$ be a separable unital strongly self-absorbing $ \mathrm {C}^*$-algebra.

  1. If $ A$ is separable and $ \mathcal {D}$-absorbing (i.e., $ A \otimes \mathcal {D} \cong A$), and $ E$ has the Rokhlin property, then $ P$ is $ \mathcal {D}$-absorbing.
  2. If $ \mathcal {D}$ is the universal UHF algebra $ \mathcal {U}_\infty $ and $ \alpha \colon G \rightarrow \mathrm {Aut}(D)$ has the Rokhlin property, then $ \mathcal {U}_\infty \rtimes _\alpha G \cong \mathcal {U}_\infty $.
  3. If $ A = \mathcal {D}$ is an inductive limit of weakly semiprojective $ \mathrm {C}^*$-algebras and $ E$ has the Rokhlin property, then $ P \cong \mathcal {D}$. In particular, this is true when $ \mathcal {D}$ is a $ \mathrm {UHF}$ algebra of infinite type, $ \mathcal {O}_2$, or $ \mathcal {O}_\infty $.
As an application, if $ A$ is a unital $ \mathcal {D}$-absorbing $ \mathrm {C}^*$-algebra and $ \alpha $ is an action from a finite group $ G$ on $ A$ with the Rokhlin property, then we have

  1. the crossed product algebra $ A \rtimes _\alpha G$ is also $ \mathcal {D}$-absorbing,
  2. for any subgroup $ H \subset G$, $ A^H$ is $ \mathcal {D}$-absorbing,
  3. if $ A$ is $ \mathcal {O}_2$, then $ A \rtimes _\alpha G \cong \mathcal {O}_2$.

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Additional Information

Hiroyuki Osaka
Affiliation: Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan
Email: osaka@se.ritsumei.ac.jp

Tamotsu Teruya
Affiliation: Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan
Address at time of publication: Faculty of Education, Gunma University, 4-2 Aramaki-machi, Maebashi City, Gunma, 371-8510, Japan
Email: teruya@se.ritsumei.ac.jp, teruya@gunma-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2013-05907-7
Received by editor(s): March 22, 2010
Received by editor(s) in revised form: May 14, 2011, October 25, 2011, March 28, 2012, and April 4, 2012
Published electronically: September 13, 2013
Additional Notes: The first author’s research was partially supported by the JSPS grant for Scientific Research No. 20540220
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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