The generator problem for $\mathcal {Z}$-stable $C^*$-algebras
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- by Hannes Thiel and Wilhelm Winter PDF
- Trans. Amer. Math. Soc. 366 (2014), 2327-2343 Request permission
Abstract:
The generator problem was posed by Kadison in 1967, and it remains open today. We provide a solution for the class of $C^*$-algebras absorbing the Jiang-Su algebra $\mathcal {Z}$ tensorially. More precisely, we show that every unital, separable, $\mathcal {Z}$-stable $C^*$-algebra $A$ is singly generated, which means that there exists an element $x\in A$ that is not contained in any proper sub-$C^*$-algebra of $A$.
To give applications of our result, we observe that $\mathcal {Z}$ can be embedded into the reduced group $C^*$-algebra of a discrete group that contains a non-cyclic, free subgroup. It follows that certain tensor products with reduced group $C^*$-algebras are singly generated. In particular, $C^*_r(F_\infty )\otimes C^*_r(F_\infty )$ is singly generated.
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Additional Information
- Hannes Thiel
- Affiliation: Mathematisches Institut der Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
- MR Author ID: 930802
- Email: hannes.thiel@uni-muenster.de
- Wilhelm Winter
- Affiliation: Mathematisches Institut der Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
- MR Author ID: 671014
- Email: wwinter@uni-muenster.de
- Received by editor(s): April 18, 2012
- Published electronically: February 3, 2014
- Additional Notes: This research was partially supported by the Centre de Recerca Matemàtica, Barcelona, and the DFG through SFB 878.
The first author was partially supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation, Copenhagen
The second author was partially supported by EPSRC Grants EP/G014019/1 and EP/I019227/1. - © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 2327-2343
- MSC (2010): Primary 46L05, 46L85; Secondary 46L35
- DOI: https://doi.org/10.1090/S0002-9947-2014-06013-3
- MathSciNet review: 3165640