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Transactions of the American Mathematical Society

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Fixed-points in the cone of traces on a $ C^{\ast}$-algebra


Author: Mikael Rørdam
Journal: Trans. Amer. Math. Soc. 371 (2019), 8879-8906
MSC (2010): Primary 46L35; Secondary 46L05, 37A55
DOI: https://doi.org/10.1090/tran/7797
Published electronically: February 28, 2019
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Abstract: Nicolas Monod introduced the class of groups with the fixed-point property for cones, characterized by always admitting a nonzero fixed-point when acting (suitably) on proper weakly complete cones. He proved that his class of groups contains the class of groups with subexponential growth and is contained in the class of supramenable groups. In this paper we investigate what Monod's results say about the existence of invariant traces on (typically nonunital) $ C^*$-algebras equipped with an action of a group with the fixed-point property for cones. As an application of these results, we provide results on the existence (and nonexistence) of traces on the (nonuniform) Roe algebra.


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Additional Information

Mikael Rørdam
Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100, Copenhagen Ø, Denmark
Email: rordam@math.ku.dk

DOI: https://doi.org/10.1090/tran/7797
Received by editor(s): August 4, 2018
Received by editor(s) in revised form: January 1, 2019
Published electronically: February 28, 2019
Additional Notes: Supported by the Danish Council for Independent Research, Natural Sciences, and the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation at the University of Copenhagen.
Dedicated: Dedicated to the memory of John Roe
Article copyright: © Copyright 2019 American Mathematical Society