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Classifiable \({\textrm{C}}^*\)-algebras from minimal \({\mathbb {Z}}\)-actions and their orbit-breaking subalgebras

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In this paper we consider the question of what abelian groups can arise as the K-theory of \({\textrm{C}}^*\)-algebras arising from minimal dynamical systems. We completely characterize the K-theory of the crossed product of a space X with finitely generated K-theory by an action of the integers and show that crossed products by a minimal homeomorphisms exhaust the range of these possible K-theories. Moreover, we may arrange that the minimal systems involved are uniquely ergodic, so that their \({\textrm{C}}^*\)-algebras are classified by their Elliott invariants. We also investigate the K-theory and the Elliott invariants of orbit-breaking algebras. We show that given arbitrary countable abelian groups \(G_0\) and \(G_1\) and any Choquet simplex \(\Delta \) with finitely many extreme points, we can find a minimal orbit-breaking relation such that the associated \({\textrm{C}}^*\)-algebra has K-theory given by this pair of groups and tracial state space affinely homeomorphic to \(\Delta \). We also improve on the second author’s previous results by using our orbit-breaking construction to \({\textrm{C}}^*\)-algebras of minimal amenable equivalence relations with real rank zero that allow torsion in both \(K_0\) and \(K_1\). These results have important applications to the Elliott classification program for \({\textrm{C}}^*\)-algebras. In particular, we make a step towards determining the range of the Elliott invariant of the \({\textrm{C}}^*\)-algebras associated to étale equivalence relations.

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Acknowledgements

The authors thank the Banff International Research Stations and the organizers of the workshop Future Targets in the Classification Program for Amenable \({\textrm{C}}^*\)-Algebras, where this project was initiated. Thanks also to the University of Victoria and University of Colorado, Boulder for research visits facilitating this collaboration. Work on the project also benefitted from the 2018 conference on Cuntz–Pimsner algebras at the Lorentz Center, which was attended by the first and third listed authors.

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Authors and Affiliations

  1. Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, CO, 80309-0395, USA

    Robin J. Deeley

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada

    Ian F. Putnam

  3. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67, Prague, Czech Republic

    Karen R. Strung

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Correspondence to Karen R. Strung.

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Robin J. Deeley is currently funded by NSF Grant DMS 2000057 and was previously funded by Simons Foundation Collaboration Grant for Mathematicians number 638449. Karen R. Strung is currently funded by GAČR project 20-17488Y and RVO: 67985840 and part of this work was carried out while funded by Sonata 9 NCN grant 2015/17/D/ST1/02529 and a Radboud Excellence Initiative Postdoctoral Fellowship. Ian F. Putnam is supported in part by an NSERC Discovery Grant.

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Deeley, R.J., Putnam, I.F. & Strung, K.R. Classifiable \({\textrm{C}}^*\)-algebras from minimal \({\mathbb {Z}}\)-actions and their orbit-breaking subalgebras. Math. Ann. 388, 703–729 (2024). https://doi.org/10.1007/s00208-022-02526-1

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