On the structure of tensor products ofℓ_p-spaces

Abstract

We examine some structural properties of (injective and projective) tensor products of ^-spaces (projections, complemented subspaces, reflexivity, isomorphisms, etc.). We combine these results with combinatorial arguments to address the question of primarity for these spaces and their duals.Introduction.

Banach spaces of homogeneous polynomials without the approximation property

“…Propositions 2.1 and 2.2 extend results of Pisier [26], page 316, and Arias and Farmer [2], page 17, respectively, and play a key role in the proofs of our main results, namely Theorems 3.1 and 3.2.…”

“…Then 1 < r p/(p − 1) and another result of Arias and Farmer [2], Theorem 1.3, implies that E contains a complemented subspace isomorphic to ℓ r . Hence it follows that…”

“…Among them we mention Szankowski's famous counterexample [27], the complementation properties of L p spaces established by Pe lczyński [23], the isomorphisms between spaces of operators on L p spaces and spaces of operators on ℓ p spaces discovered by Arias and Farmer [2], the isomorphisms between spaces of multilinear forms and spaces of symmetric multilinear forms on stable Banach spaces discovered by Diaz and Dineen [7], the complementation properties of spaces of homogeneous polynomials obtained by Aron and Schottenloher [3], and the complementation properties of tensor products of ℓ p spaces, obtained also by Arias and Farmer [2]. These results play a key role in our proofs, and some of them are applied several times.…”

On the simplicial cohomology of Banach operator algebras

“…σ m : X ⊗ A ⊗n → X ⊗ A ⊗n ) by σ m (a 1 ⊗ · · · ⊗ a n ) := s m a 1 r m ⊗ · · · ⊗ s m a n r m (2) resp. σ m (x ⊗ a 1 ⊗ · · · ⊗ a n ) := s m xr m ⊗ s m a 1 r m ⊗ · · · ⊗ s m a n r m .…”

Block diagonal polynomials

“…The space of diagonal polynomials can be seen as the dual of the subspace of k π,s E (the complete k-fold symmetric tensor product of E endowed with the projective norm) spanned by the tensor diagonal basis {e n ⊗ · · · ⊗ e n } n∈N . Some aspects of the tensor diagonal basis for non-symmetric tensor products were studied by Holub [21] and Arias and Farmer [1].…”