  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.
