On isometry groups and maximal symmetry

Abstract

We study problems of maximal symmetry in Banach spaces. This is done by providing an analysis of the structure of small subgroups of the general linear group $\mathrm{GL}\left(X\right)$, where $X$ is a separable reflexive Banach space. In particular, we provide the first known example of a Banach space $X$ without any equivalent maximal norm, or equivalently such that $\mathrm{GL}\left(X\right)$ contains no maximal bounded subgroup. Moreover, this space $X$ may be chosen to be super-reflexive.