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

Abstract

We prove that the spaces ℓp, 1<p<∞, p≠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 ℓ22 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 ℓp, 1<p<∞, p≠2, have continuum different renormings with 1-unconditional bases each with a different maximal isometry group, and that every symmetric space other than ℓ2 has at least a countable number of such renormings. On the other hand we show that the spaces ℓp, 1<p<∞, p≠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 ℓp.