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

TL;DR

This paper investigates tensor norms that eliminate unconditional bases in tensor products of Banach spaces, providing criteria and showing most common tensor norms destroy unconditionality, with applications to polynomial and operator ideals.

Contribution

It introduces a simple criterion to determine when tensor norms destroy unconditionality and applies it to show most injective and projective norms do so, impacting polynomial and operator ideal theory.

Findings

Most injective and projective tensor norms destroy unconditionality.

Many polynomial ideals lack the Gordon-Lewis property.

The unconditionality of monomial basic sequences is analyzed.

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 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.