On an isomorphic Banach-Mazur rotation problem and maximal norms in Banach spaces

TL;DR

This paper advances the understanding of Banach space geometry by showing certain spaces lack almost transitive renormings, exploring asymptotic structures, and constructing diverse renormings with distinct isometry groups.

Contribution

It proves that $ ext{ell}_p$ spaces for $p eq 2$ and their subspaces do not admit almost transitive renormings, and constructs many renormings with unique isometry groups, addressing the Banach-Mazur rotation problem.

Findings

Spaces $ ext{ell}_p$, $p eq 2$, lack almost transitive renormings.

Constructed continuum many renormings with distinct maximal isometry groups.

Identified properties of asymptotic structures in almost transitive spaces.

Abstract

We prove that the spaces $\ell_p$, $1<p<\infty, p\ne 2$, and all infinite-dimensional subspaces of their quotient spaces do not admit equivalent almost transitive renormings. This is a step towards the solution of the Banach-Mazur rotation problem, which asks whether a separable Banach space with a transitive norm has to be isometric or isomorphic to a Hilbert space. We obtain this as a consequence of a new property of almost transitive spaces with a Schauder basis, namely we prove that in such spaces the unit vector basis of $\ell_2^2$ belongs to the two-dimensional asymptotic structure and we obtain some information about the asymptotic structure in higher dimensions. Further, we prove that the spaces $\ell_p$, $1<p<\infty$, $p\ne 2$, have continuum different renormings with 1-unconditional bases each with a different maximal isometry group, and that every symmetric space other than $\ell_2$ has at least a countable number of such renormings. On the other hand we show that the spaces $\ell_p$, $1<p<\infty$, $p\ne 2$, have continuum different renormings each with an isometry group which is not contained in any maximal bounded subgroup of the group of isomorphisms of $\ell_p$.