A fixed point theorem in $B(H,\ell _{\infty })$

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