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

TL;DR

This paper proves the existence of proper invariant metrics on locally compact second countable groups and extends results on proper affine actions to reflexive Banach spaces, enhancing understanding of group actions on Banach spaces.

Contribution

It establishes proper invariant metrics on such groups and extends proper affine action results to reflexive Banach spaces, broadening the scope of group action theory.

Findings

Existence of proper invariant metrics on locally compact second countable groups.

Extension of proper affine actions to reflexive Banach spaces.

Every such group admits a proper affine isometric action on a specific Banach space.

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