Extending polynomials in maximal and minimal ideals

Given an homogeneous polynomial on a Banach space $E$ belonging to some maximal or minimal polynomial ideal, we consider its iterated extension to an ultrapower of $E$ and prove that this extension remains in the ideal and has the same ideal norm. As a consequence, we show that the Aron-Berner extension is a well defined isometry for any maximal or minimal ideal of homogeneous polynomials. This allow us to obtain symmetric versions of some basic results of the metric theory of tensor products.Comment: 13 page

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

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 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 $\varepsilon$ and $\pi$ 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