This is an announcement for the paper "Unconditionality in tensor products and ideals of polynomials, multilinear forms and operators" by Daniel Carando and Daniel Galicer.
Abstract: We study tensor norms that destroy unconditionality in the following sense: for every Banach space $E$ with unconditional basis, the $n$-fold tensor product of $E$ (with the corresponding tensor norms) does not have unconditional basis. We show that this holds for all injective and projective tensor norms different from $\varepsilon$ and $\pi$, both in the full and symmetric tensor products. In particular, every nontrivial natural symmetric tensor norms destroys unconditionality. We prove that there are exactly 6 natural symmetric tensor norms for $n\ge 3$, a noteworthy difference with the 2-fold case. We present applications to polynomial ideals: we show that many polynomial ideals never have the Gordon-Lewis property or, in the spirit of a result of Defant and Kalton, can have the Gordon-Lewis property but never have unconditional basis. We also consider unconditionality in multilinear and operator ideals.
Archive classification: math.FA
Mathematics Subject Classification: 46M05; 46G25; 47L20
Remarks: 27 pages
The source file(s), Carando-GalicerArxiv.tex: 100018 bytes, is(are) stored in gzipped form as 0906.3253.gz with size 26kb. The corresponding postcript file has gzipped size 163kb.
Submitted from: dgalicer@dm.uba.ar
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