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The weak Haagerup property


Author: Søren Knudby
Journal: Trans. Amer. Math. Soc. 368 (2016), 3469-3508
MSC (2010): Primary 22D25; Secondary 22D15
DOI: https://doi.org/10.1090/tran/6445
Published electronically: August 19, 2015
MathSciNet review: 3451883
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Abstract: We introduce the weak Haagerup property for locally compact groups and prove several hereditary results for the class of groups with this approximation property. The class contains a priori all weakly amenable groups and groups with the usual Haagerup property, but examples are given of groups with the weak Haagerup property which are not weakly amenable and do not have the Haagerup property.

In the second part of the paper we introduce the weak Haagerup property for finite von Neumann algebras, and we prove several hereditary results here as well. Also, a discrete group has the weak Haagerup property if and only if its group von Neumann algebra does.

Finally, we give an example of two $ \mathrm {II}_1$ factors with different weak Haagerup constants.


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Additional Information

Søren Knudby
Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitets- parken 5, DK-2100 Copenhagen Ø, Denmark
Address at time of publication: Mathematical Institute, University of Münster, Einsteinstrasse 62, 48149 Münster, Germany
Email: knudby@math.ku.dk, knudby@uni-muenster.de

DOI: https://doi.org/10.1090/tran/6445
Received by editor(s): February 5, 2014
Received by editor(s) in revised form: March 18, 2014
Published electronically: August 19, 2015
Additional Notes: The author was supported by ERC Advanced Grant No. OAFPG 247321 and the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).
Article copyright: © Copyright 2015 American Mathematical Society