Open Access
February 2010 Fixed point-free isometric actions of topological groups on Banach spaces
Lionel Nguyen Van Thé, Vladimir G. Pestov
Bull. Belg. Math. Soc. Simon Stevin 17(1): 29-51 (February 2010). DOI: 10.36045/bbms/1267798497

Abstract

We show that every non-precompact topological group admits a fixed point-free continuous action by affine isometries on a suitable Banach space. Thus, precompact groups are defined by the fixed point property for affine isometric actions on Banach spaces. For separable topological groups, in the above statements it is enough to consider affine actions on one particular Banach space: the unique Banach space envelope $\langle\mathbb U\rangle$ of the universal Urysohn metric space $\mathbb U$, known as the Holmes space. At the same time, we show that Polish groups need not admit topologically proper (in particular, free) affine isometric actions on Banach spaces (nor even on complete metric spaces): this is the case for the unitary group $U(\ell^2)$ with strong operator topology, the infinite symmetric group $S_\infty$, etc.

Citation

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Lionel Nguyen Van Thé. Vladimir G. Pestov. "Fixed point-free isometric actions of topological groups on Banach spaces." Bull. Belg. Math. Soc. Simon Stevin 17 (1) 29 - 51, February 2010. https://doi.org/10.36045/bbms/1267798497

Information

Published: February 2010
First available in Project Euclid: 5 March 2010

zbMATH: 1207.22002
MathSciNet: MR2656670
Digital Object Identifier: 10.36045/bbms/1267798497

Subjects:
Primary: 22A25 , ‎43A65 , 57S99

Keywords: Affine isometric actions , fixed point-free actions , free isometric actions , Holmes space , precompact groups , proper actions , Property (FH) , Urysohn metric space

Rights: Copyright © 2010 The Belgian Mathematical Society

Vol.17 • No. 1 • February 2010
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