On isometry groups and maximal symmetry

TL;DR

This paper investigates the structure of symmetry groups in Banach spaces, providing the first example of a space without an equivalent maximal norm and analyzing the implications for maximal symmetry.

Contribution

It introduces the first example of a Banach space with no equivalent maximal norm, advancing understanding of symmetry group structures in Banach spaces.

Findings

Constructed a super-reflexive Banach space with no maximal bounded subgroup in GL(X)

Analyzed the structure of small subgroups of GL(X) in separable reflexive Banach spaces

Provided new insights into the maximal symmetry properties of Banach spaces

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 GL(X), 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 GL(X) contains no maximal bounded subgroup. Moreover, this space X may be chosen to be super-reflexive.