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

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dc.creatorVan Thé, Lionel Nguyen
dc.creatorPestov, Vladimir G.
dc.date2008-04-09
dc.date2008-11-06
dc.date.accessioned2026-07-07T10:15:33Z
dc.date.available2026-07-07T10:15:33Z
dc.descriptionWe 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 of the universal Urysohn metric space, 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 of the separable infinite dimensional Hilbert space with strong operator topology, the infinite symmetric group, etc.
dc.description20 pages, to appear in the Bulletin of the Belgian Mathematical Society, referee's comments incorporated
dc.identifierhttps://arxiv.org/abs/0804.1583
dc.identifierhttp://arxiv.org/abs/0804.1583
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/173210
dc.subjectGroup Theory
dc.subject22A25; 43A65; 57S99
dc.titleFixed point-free isometric actions of topological groups on Banach spaces
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