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

1310.7139Oct 26, 2013small editorial and stylistic changes

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

S. J. Dilworth, B. Randrianantoanina

Functional Analysis

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We prove that the spaces $\ell_p$ , $1<p<\infty, p\ne 2$

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, 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

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