This is an announcement for the paper "Randomized series and geometry of Banach spaces" by Han Ju Lee.

Abstract: We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it is representable in the Lebesgue-Bochner $L_p(X)$. New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice $E$ is uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is uniformly convex and that a K"othe function space $E$ is upper locally uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is midpoint locally uniformly convex.

Archive classification: math.FA

Mathematics Subject Classification: 46B20;46B07;46B09

The source file(s), randomized-series2007-01-29.tex: 33940 bytes, is(are) stored in gzipped form as 0706.3740.gz with size 10kb. The corresponding postcript file has gzipped size 96kb.

Submitted from: hahnju@postech.ac.kr

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