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

TL;DR

This paper proves a fixed point theorem for isometry groups in certain metric spaces, with applications to injective spaces and derivations in Banach modules, extending known results in functional analysis.

Contribution

It establishes a new fixed point theorem for isometry groups in metric spaces with uniform relative normal structure and applies it to problems in Banach space theory.

Findings

Existence of fixed points for isometry groups with bounded orbits

Extension of Lang's theorem to injective metric spaces

Inner derivations in Banach $L_1$-bimodules are characterized

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.