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Ranks of Operators in Simple \(C^*\)-algebras with Stable Rank One

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Abstract

Let A be a simple \(C^*\)-algebra with stable rank one. We show that every strictly positive, lower semicontinuous, affine function on the simplex of normalized quasitraces of A is realized as the rank of an operator in the stabilization of A. Assuming moreover that A has locally finite nuclear dimension, we deduce that A is \(\mathcal {Z}\)-stable if and only if it has strict comparison of positive elements. In particular, the Toms–Winter conjecture holds for simple, approximately subhomogeneous \(C^*\)-algebras with stable rank one.

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Acknowledgements

I thank Nate Brown, Tim de Laat, Leonel Robert, Stuart White and Wilhelm Winter for valuable comments. I am grateful to Eusebio Gardella for his feedback on the first draft of this paper. Further, I want to thank the anonymous referee for many helpful suggestions.

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  1. Mathematisches Institut, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62, 48149, Münster, Germany

    Hannes Thiel

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Communicated by Y. Kawahigashi

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The author was partially supported by the Deutsche Forschungsgemeinschaft (SFB 878 Groups, Geometry & Actions).

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Thiel, H. Ranks of Operators in Simple \(C^*\)-algebras with Stable Rank One. Commun. Math. Phys. 377, 37–76 (2020). https://doi.org/10.1007/s00220-019-03491-8

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