| Popis: |
We prove that the spaces l p , 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 l 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 l p , 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 l 2 has at least a countable number of such renormings. On the other hand we show that the spaces l p , 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 l p . |
Full Text from ScienceDirect