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