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

Abstract

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 ${⨁}_{n=1}^{\infty }{L}^{2n}(G,d\mu ),$ where the direct sum is taken in the $l^2$-sense.

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