Strongly self-absorbing $C^{*}$-algebras
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- by Andrew S. Toms and Wilhelm Winter PDF
- Trans. Amer. Math. Soc. 359 (2007), 3999-4029 Request permission
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.References
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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
- Email: atoms@mathstat.yorku.ca
- Wilhelm Winter
- Affiliation: Mathematisches Institut der Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany
- MR Author ID: 671014
- Email: wwinter@math.uni-muenster.de
- 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).
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 3999-4029
- MSC (2000): Primary 46L85, 46L35
- DOI: https://doi.org/10.1090/S0002-9947-07-04173-6
- MathSciNet review: 2302521
Dedicated: Dedicated to George Elliott on the occasion of his 60th birthday.