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 , where is a separable reflexive Banach space. In particular, we provide the first known example of a Banach space without any equivalent maximal norm, or equivalently such that contains no maximal bounded subgroup. Moreover, this space may be chosen to be super-reflexive.
"On isometry groups and maximal symmetry." Duke Math. J. 162 (10) 1771 - 1831, 15 July 2013. https://doi.org/10.1215/00127094-2322898