Published by De Gruyter February 8, 2016

Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang–Su algebra

Robin J. Deeley , Ian F. Putnam and Karen R. Strung

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2015-0091

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Abstract

The principal aim of the present paper is to give a dynamical presentation of the Jiang–Su algebra. Originally constructed as an inductive limit of prime dimension drop algebras, the Jiang–Su algebra has gone from being a poorly understood oddity to having a prominent positive role in George Elliott’s classification programme for separable, nuclear C*-algebras. Here, we exhibit an étale equivalence relation whose groupoid C*-algebra is isomorphic to the Jiang–Su algebra. The main ingredient is the construction of minimal homeomorphisms on infinite, compact metric spaces, each having the same cohomology as a point. This construction is also of interest in dynamical systems. Any self-map of an infinite, compact space with the same cohomology as a point has Lefschetz number one. Thus, if such a space were also to satisfy some regularity hypothesis (which our examples do not), then the Lefschetz–Hopf Theorem would imply that it does not admit a minimal homeomorphism.

Funding statement: The first author was supported by ANR Project SingStar. The second author was supported by an NSERC Discovery Grant.

Acknowledgements

The authors thank the Banff International Research Station and the organisers of the workshop “Dynamics and C*-Algebras: Amenability and Soficity”, where this project was initiated. In particular, this work followed a suggestion of Wilhelm Winter, for which the authors are indebted. The authors thank the Department of Mathematics and Statistics at the University of Victoria and the Banach Center, Institute of Mathematics of the Polish Academy of Sciences for funding research visits facilitating this collaboration. The third author thanks Stuart White for useful discussions. Further thanks go to Magnus Goffeng and Adam Skalski for proofreading an initial draft. Finally, thanks to Alcides Buss, Xin Li and Aidan Sims for pointing out Remarks 2.6 (ii) and 2.10 following a talk on this work at the “Workshop on C*-algebras: Groups and Actions” at the University of Münster.

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Received: 2015-03-19

Revised: 2015-09-08

Published Online: 2016-02-08

Published in Print: 2018-09-01

Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang–Su algebra

Abstract

Acknowledgements

References

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