On the structure of tensor products ofℓp-spaces

Abstract

A Banach space X is prime if every infinite-dimensional complemented subspace contains a further subspace which is isomorphic to X. A Banach space X is said to be primary if whenever X — Y © Z, X is isomorphic to either Y or Z. The classical examples of prime spaces are the spaces ipj 1 < p < oc. Many spaces derived from the £p-spaces in various ways are primary (see for example [AEO] and [CL]). The primarity of B(H) was shown by Blower [B] in 1990, and Arias [A] has recently developed further techniques which are used to prove the primarity of Cι, the space of trace class operators (this was first shown by Arazy [Arl, Ar2]). It has become clear that these techniques are not naturally confined to a Hubert space context; in the present paper we wish to extend the results to a variety of tensor products and operator spaces of ^-spaces (and in some cases £p-spaces). We also include some related results. Some of the intermediate propositions (on factoring operators through the identity) may actually be true for a wider class of Banach spaces (those with unconditional bases which have nontrivial lower and upper estimates). In fact, the combinatorial aspects of the factorization can be applied quite generally, and may have other applications. The proofs of primarity, however, rely on Pelczyήski's decomposition method which is not so readily extended. We have thus kept mainly to the case of injective and projective tensor products of tv spaces throughout. The results we obtain apply to the growing study of polynomials on Banach spaces since polynomials may be considered as symmetric multilinear operators with an equivalent norm (see [FJ], [M],