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

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 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.

arXiv comment20 pages, to appear in the Bulletin of the Belgian Mathematical Society, referee's comments incorporated

References in corpus (5)

• Isometric group actions on Banach spaces and representations vanishing at infinity
• Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces
• Linearizability of Non-expansive Semigroup Actions on Metric Spaces
• A topological version of the Bergman property
• The Urysohn sphere is oscillation stable