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

DANIEL CARANDO; DANIEL GALICER

QUARTERLY JOURNAL OF MATHEMATICS

OXFORD UNIV PRESS

Año: 2011 vol. 62 p. 845 - 869

0033-5606

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 $arepsilon$ 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. In some cases we even obtain that the monomial basic sequence can never be unconditional. Analogous problems for multilinear ideals are addressed, and noteworthy differences between the 2-fold and the $n$-fold ($nge 3$) theory are obtained.