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 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.

References in corpus (3)

• Property (T) and rigidity for actions on Banach spaces
• The Dixmier problem, lamplighters and Burnside groups
• Non-unitarisable representations and random forests

Cited by in corpus (2)

• Light groups of isomorphisms of Banach spaces and invariant LUR renormings
• Non-unitarisable representations and maximal symmetry