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Fixed point property and the Fourier algebra of a locally compact group


Authors: Anthony To-Ming Lau and Michael Leinert
Journal: Trans. Amer. Math. Soc. 360 (2008), 6389-6402
MSC (2000): Primary 43A15, 47A09; Secondary 43A20, 47H10, 46B22
DOI: https://doi.org/10.1090/S0002-9947-08-04622-9
Published electronically: July 22, 2008
MathSciNet review: 2434292
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Abstract: We establish some characterizations of the weak fixed point property (weak fpp) for noncommutative (and commutative) $ \mathcal{L}^1$ spaces and use this for the Fourier algebra $ A(G)$ of a locally compact group $ G.$ In particular we show that if $ G$ is an IN-group, then $ A(G)$ has the weak fpp if and only if $ G$ is compact. We also show that if $ G$ is any locally compact group, then $ A(G)$ has the fixed point property (fpp) if and only if $ G$ is finite. Furthermore if a nonzero closed ideal of $ A(G)$ has the fpp, then $ G$ must be discrete.


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Additional Information

Anthony To-Ming Lau
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: tlau@math.ualberta.ca

Michael Leinert
Affiliation: Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld, Gebäude 294, 69120 Heidelberg, Germany
Email: leinert@math.uni-heidelberg.de

DOI: https://doi.org/10.1090/S0002-9947-08-04622-9
Keywords: Weak fixed point property, nonexpansive mapping, Fourier algebra, noncommutative $\mathcal {L}^1$ space, semifinite von~Neumann algebra
Received by editor(s): November 10, 2006
Published electronically: July 22, 2008
Additional Notes: The research of the first author was supported by NSERC Grant A-7679
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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