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Nuclear dimension, \(\mathcal{Z }\)-stability, and algebraic simplicity for stably projectionless \(C^*\)-algebras

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The main result here is that a simple separable \(C^*\)-algebra is \(\mathcal{Z }\)-stable (where \(\mathcal{Z }\) denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the main results of Toms (Invent Math 183(2):225–244, 2011) and Winter (Invent Math 187(2):259–342, 2012) to the nonunital setting. As a consequence, finite nuclear dimension implies \(\mathcal{Z }\)-stability even in the case of a separable \(C^*\)-algebra with finitely many ideals. Algebraic simplicity is established as a fruitful weakening of being simple and unital, and the proof of the main result makes heavy use of this concept.

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Acknowledgments

The author would like to thank Joachim Cuntz, George Elliott, Dominic Enders, Ilijas Farah, Bhishan Jacelon, Leonel Robert, Luis Santiago, Ján Špakula, Hannes Thiel, Andrew Toms, Stuart White, and Wilhelm Winter for discussions promoting the research in this article.

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  1. Institute of Mathematics, University of Aberdeen, Aberdeen, AB24 3UE, UK

    Aaron Tikuisis

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Correspondence to Aaron Tikuisis.

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This research was supported by DFG (SFB 878).

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Tikuisis, A. Nuclear dimension, \(\mathcal{Z }\)-stability, and algebraic simplicity for stably projectionless \(C^*\)-algebras. Math. Ann. 358, 729–778 (2014). https://doi.org/10.1007/s00208-013-0951-0

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