DOI: 10.1515/forum-2023-0193 ISSN: 0933-7741

A fixed point theorem for isometries on a metric space

Andrzej Wiśnicki
  • Applied Mathematics
  • General Mathematics

Abstract

We show that if X is a complete metric space with uniform relative normal structure and G is a subgroup of the isometry group of X with bounded orbits, then there is a point in X fixed by every isometry in G. As a corollary, we obtain a theorem of U. Lang (2013) concerning injective metric spaces. A few applications of this theorem are given to the problems of inner derivations. In particular, we show that if

L 1 ( μ ) {L_{1}(\mu)}
is an essential Banach
L 1 ( G ) {L_{1}(G)}
-bimodule, then any continuous derivation
δ : L 1 ( G ) L ( μ ) {\delta:L_{1}(G)\rightarrow L_{\infty}(\mu)}
is inner. This extends a theorem of B. E. Johnson (1991) asserting that the convolution algebra
L 1 ( G ) {L_{1}(G)}
is weakly amenable if G is a locally compact group.

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