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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Strongly self-absorbing $ C^{*}$-algebras

Authors: Andrew S. Toms and Wilhelm Winter
Journal: Trans. Amer. Math. Soc. 359 (2007), 3999-4029
MSC (2000): Primary 46L85, 46L35
Published electronically: March 20, 2007
MathSciNet review: 2302521
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Abstract: Say that a separable, unital $ C^*$-algebra $ \mathcal{D} \ncong \mathbb{C}$ is strongly self-absorbing if there exists an isomorphism $ \varphi: \mathcal{D} \to \mathcal{D} \otimes \mathcal{D}$ such that $ \varphi$ and $ \mathrm{id}_{\mathcal{D}} \otimes \mathbf{1}_{\mathcal{D}}$ are approximately unitarily equivalent $ *$-homomorphisms. We study this class of algebras, which includes the Cuntz algebras $ \mathcal{O}_2$, $ \mathcal{O}_{\infty}$, the UHF algebras of infinite type, the Jiang-Su algebra $ \mathcal{Z}$ and tensor products of $ \mathcal{O}_{\infty}$ with UHF algebras of infinite type. Given a strongly self-absorbing $ C^{*}$-algebra $ \mathcal{D}$ we characterise when a separable $ C^*$-algebra absorbs $ \mathcal{D}$ tensorially (i.e., is $ \mathcal{D}$-stable), and prove closure properties for the class of separable $ \mathcal{D}$-stable $ C^*$-algebras. Finally, we compute the possible $ K$-groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing $ C^*$-algebras.

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Additional Information

Andrew S. Toms
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3

Wilhelm Winter
Affiliation: Mathematisches Institut der Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany

Keywords: Nuclear $C^*$-algebras, K-theory, classification
Received by editor(s): March 28, 2005
Received by editor(s) in revised form: August 15, 2005
Published electronically: March 20, 2007
Additional Notes: The first author was supported by an NSERC Postdoctoral Fellowship, and the second author by DFG (through the SFB 478), EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280).
Dedicated: Dedicated to George Elliott on the occasion of his 60th birthday.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.