Universal proper G-spaces

Abstract

It is proved that: (1) every Lie group G can act properly (in sense of Palais) on each infinite-dimensional Hilbert space 12 (T) of a given weight tau such that (G, l(2) (tau)) becomes a universal G-space for all metrizable proper G-spaces admitting an invariant metric and having weight less than or equal to tau

(2) every Lie group G can act properly on R-tau \ {0} such that (G, R-tau \ {0}) becomes a universal G-space for all Tychonoff proper G-spaces of weight less than or equal to tau

(3) there is a dispersive dynamical system on l(2), universal for all separable, metrizable, dispersive dynamical systems having a regular orbit space. Other universal proper G-spaces are constructed. As a corollary a shorter proof of Palais' invariant metric existence theorem is obtained. The metric cones con(G/H), with H c G a compact Subgroup, are the main building blocs in our approach. (C) 2002 Elsevier Science B.V. All rights reserved.