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Unconditionality in tensor products and ideals of polynomials, multilinear forms and operators

Abstract

We study tensor norms that destroy unconditionality in the following sense: for every Banach space EE with unconditional basis, the nn-fold tensor product of EE (with the corresponding tensor norm) does not have unconditional basis. We establish an easy criterion to check weather a tensor norm destroys unconditionality or not. Using this test we get that all injective and projective tensor norms different from ε\varepsilon and π\pi destroy unconditionality, both in full and symmetric tensor products. We present applications to polynomial ideals: we show that many usual polynomial ideals never enjoy the Gordon-Lewis property. We also consider the unconditionality of the monomial basic sequence. Analogous problems for multilinear and operator ideals are addressed.Comment: 23 page

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