Bulletin of the Belgian Mathematical Society - Simon Stevin

Fixed point-free isometric actions of topological groups on Banach spaces

Lionel Nguyen Van Thé and Vladimir G. Pestov

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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.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 17, Number 1 (2010), 29-51.

First available in Project Euclid: 5 March 2010

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Zentralblatt MATH identifier

Primary: 22A25: Representations of general topological groups and semigroups 43A65: Representations of groups, semigroups, etc. [See also 22A10, 22A20, 22Dxx, 22E45] 57S99: None of the above, but in this section

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


Nguyen Van Thé, Lionel; Pestov, Vladimir G. Fixed point-free isometric actions of topological groups on Banach spaces. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 1, 29--51. doi:10.36045/bbms/1267798497. https://projecteuclid.org/euclid.bbms/1267798497

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