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The generator problem for $ \mathcal{Z}$-stable $ C^*$-algebras


Authors: Hannes Thiel and Wilhelm Winter
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
Published electronically: February 3, 2014
MathSciNet review: 3165640
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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
Email: hannes.thiel@uni-muenster.de

Wilhelm Winter
Affiliation: Mathematisches Institut der Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
Email: wwinter@uni-muenster.de

DOI: https://doi.org/10.1090/S0002-9947-2014-06013-3
Keywords: $C^*$-algebras, generator problem, single generation, $\mathcal{Z}$-stability
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.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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