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

摘要

We prove that the spaces l(p), 1 < p < infinity, p not equal 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 4 belongs to the two-dimensional asymptotic structure and we obtain some information about the asymptotic structure in higher dimensions.