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TheBanach-Mazur compacta are absolute retracts

Antonian S.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 2

113-119

We prove that the space N(E) of all norms on n-dimensional linear space E is a proper G-space under the natural action of the full linear group G = GL(n). The Banach-Mazur compactum Q(n) is just the orbit space of the G-space N(E). We show here that for any n [is greater than or equal to] 1, Q(n) is an absolute retract for metric spaces. We generalize this fact be showing that for a locally compact almost connected group G, its maximal compact subgroup K, and a proper G-space X, X/G is an A(N)R space provided it is metrizable and X is a K-A(N)R space.