A fixed point theorem for isometries on a metric space

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.