This is an announcement for the paper "Convexity and smoothness of Banach spaces with numerical index one" by Vladimir Kadets, Miguel Martin, Javier Meri, and Rafael Paya .
Abstract: We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm is neither smooth nor strictly convex. Actually, these results also hold if the space has the (strictly weaker) alternative Daugavet property. We construct a (non-complete) strictly convex predual of an infinite-dimensional $L_1$ space (which satisfies a property called lushness which implies numerical index~$1$). On the other hand, we show that a lush real Banach space is neither strictly convex nor smooth, unless it is one-dimensional. In particular, if a subspace $X$ of the real space $C[0,1]$ is smooth or strictly convex, then $C[0,1]/X$ contains a copy of $C[0,1]$. Finally, we prove that the dual of any lush infinite-dimensional real space contains a copy of $\ell_1$.
Archive classification: math.FA math.OA
Mathematics Subject Classification: 46B04, 46B20, 47A12
Remarks: Illinois J. Math. (to appear)
The source file(s), Kadets-Martin-Meri-Paya.tex: 61549 bytes, is(are) stored in gzipped form as 0811.0808.gz with size 19kb. The corresponding postcript file has gzipped size 120kb.
Submitted from: mmartins@ugr.es
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