Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces

Haagerup, Uffe; Przybyszewska, Agata

Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces

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Description

In this article it is proved, that every locally compact second countable group has a left invariant metric d, which generates the topology on G, and which is proper, ie. every closed d-bounded set in G is compact. Moreover, we obtain the following extension of a result due to N. Brown and E. Guentner: Every locally compact second countable $G$ admits a proper affine action on the reflexive and strictly convex Banach space $\bigoplus^{\infty}_{n=1} L^{2n}(G, dμ),$ where the direct sum is taken in the $l^2$-sense.