Abstract of a paper by S. J. Dilworth and B. Randrianantoanina
This is an announcement for the paper "Almost transitive and maximal norms in Banach spaces" by S. J. Dilworth and B. Randrianantoanina. 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 answers a problem posed by Deville, Godefroy and Zizler in 1993. 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. We also obtain several other results about properties of classical, Tsirelson type and non-commutative Banach spaces with almost transitive norms. 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 $\ell_p$. This answers a question of Wood. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B03, 22F50 Submitted from: randrib@miamioh.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.7139 or http://arXiv.org/abs/1310.7139
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