Tensor Products Of Banach Lattices Research Articles

The positive Schur property on positive projective tensor products and spaces of regular multilinear operators

We characterize the positive Schur property in the positive projective tensor products of Banach lattices, we establish the connection with the weak operator topology and we give necessary and sufficient conditions for the space of regular multilinear operators/homogeneous polynomials taking values in a Dedekind complete Banach lattice to have the positive Schur property.

Diagonals of injective tensor products of Banach lattices with bases

Let E be a Banach lattice with a 1-unconditional basis \(\{e_i: i \in \mathbb {N}\}\). Denote by \(\Delta (\check{\otimes }_{n,\epsilon }E)\) (resp. \(\Delta (\check{\otimes }_{n,s,\epsilon }E)\)) the main diagonal space of the n-fold full (resp. symmetric) injective Banach space tensor product, and denote by \(\Delta (\check{\otimes }_{n,|\epsilon |}E)\) (resp. \(\Delta (\check{\otimes }_{n,s,|\epsilon |}E)\)) the main diagonal space of the n-fold full (resp. symmetric) injective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic. We also show that the tensor diagonal \(\{e_i\otimes \cdots \otimes e_i: i \in \mathbb {N}\}\) is a 1-unconditional basic sequence in both \(\check{\otimes }_{n,\epsilon }E\) and \(\check{\otimes }_{n,s,\epsilon }E\).

Representations of set-valued risk measures defined on the $$l$$ l -tensor product of Banach lattices

We obtain a representation for set-valued risk measures which are defined on the completed \(l\)-tensor product \(E\widetilde{\otimes }_l G\) of Banach lattices \(E\) and \(G\). This representation extends known representations for set-valued risk measures defined on Bochner spaces \(L^p(\mathbb {P}, \mathbb {R}^d)\) of \(p\)-integrable functions with values in \(\mathbb {R}^d\).

Schauder decompositions and the Fremlin projective tensor product of Banach lattices

In this paper, we first introduce a lattice decomposition and finite-dimensional lattice decomposition (FDLD) for Banach lattices. Then we show that for a Banach lattice with FDLD, the following are equivalent: (i) it has the Radon–Nikodym property; (ii) it is a KB-space; (iii) it is a Levi space; and (iv) it is a σ-Levi space. We then give a sequential representation of the Fremlin projective tensor product of an atomic Banach lattice with a Banach lattice. Using this sequential representation, we show that if one of the Banach lattices X and Y is atomic, then the Fremlin projective tensor product X⊗ˆFY has the Radon–Nikodym property (or, respectively, is a KB-space) if and only if both X and Y have the Radon–Nikodym property (or, respectively, are KB-spaces).

The Radon-Nikodym Property for Tensor Products of Banach Lattices II

The Radon-Nikodym Property for Tensor Products of Banach Lattices II

Fremlin tensor products of Banach lattices

We prove that the Fremlin tensor product of two Banach lattices with the Radon-Nikodym property, with one of them atomic, has the Radon-Nikodym property.

The Radon–Nikodym Property for Tensor Products of Banach Lattices

Let 1 p< ∞. We show thatp ˆ ⊗F X, the Fremlin projective tensor product ofp with a Banach lattice X, has the Radon-Nikodym property if and only if X has the Radon-Nikodym property; and thatp ˜ ⊗iX, the Wittstock injective tensor product of � p with a Banach lattice X, has the Radon-Nikodym property if and only if X has the Radon-Nikodym property and each positive operator frompto X is compact, where 1/p + 1/p � = 1 and letp� = c0 if p = 1. Mathematics Subject Classification 2000: 46B22, 46B42, 46M05, 47B65

Riesz Reasonable Cross Norms on Tensor Products of Banach Lattices

If E and F are Banach lattices and α is any reasonable cross norm on E ⊗ F, then there exists a reasonable cross norm | α | on E ⊗ F such that E ⊗ |α | F is a Banach lattice with respect to the ordering induced by the | α | -closure of the projective cone of E ⊗ F. The norm |α| provides a link between the Fremlin tensor product and the Wittstock normed ordered tensor product of E ⊗ F. Properties of |α|, including duals and transposes are considered. Applications to Banach latticevalued sequence spaces are given.

Bounded Variation and Tensor Products of Banach Lattices

We introduce bilinear maps of order bounded variation, semivariation and norm bounded variation. We use these notions to extend the knowledge of the projective tensor product of Banach lattices.

Factorization of positive multilinear maps

For an excellent survey of these theorems and the many applications of them we refer to J. E. Gilbert’s paper [2]. In this same article Gilbert indicates that there are available versions of weak-type factorizations for maximal operators defined by multilinear operators, but it was also noticed that strong-type factorizations for multilinear operators had not yet been studied. In this paper we shall prove strong-type factorizations for positive multilinear operators. Our approach uses the positive projective tensor product of Banach lattices and we also use some of the linear operator results of Nikisin and Maurey. The results for bilinear operators are typical for the multilinear case, but we could not restrict ourselves to the bilinear case. To prove Theorem 3.2 and Theorem 3.5 for bilinear operators with values in Lr with r > 0, we need the result of the same theorems for trilinear operators. Therefore we consider the general multilinear case. For the same reason we shall consider tensor products of n Banach lattices. The organization of this paper is as follows. In Section 2 we develop the necessary machinery of the theory of tensor products of Banach lattices. In Section 3 we prove the factorization theorems for positive multilinear operators from Lpl x... x L,,---, Lq where q>_O.

The ideal property of tensor products of Banach lattices with applications to the local structure of spaces of absolutely summing operators

The ideal property of tensor products of Banach lattices with applications to the local structure of spaces of absolutely summing operators

Order bounded operators and tensor products of Banach lattices.

Order bounded operators and tensor products of Banach lattices.

On tensor products of Banach lattices

Using techniques from harmonic analysis (more specifically Varopoulos' theory of tensor algebras) we study some tensor products of Banach lattices. Our main result is that if the Banach latticeB satisfies a certain additional property, the “Gillespie factorization property”, then a natural quotient of the spaceB⊗^B is anLl-space, and we identify the kernel of this quotient map.

Tensor products of Banach lattices

Tensor products of Banach lattices

Normed tensor products of Banach lattices

On the tensor productE⊗F of a pair of order complete Banach lattices, two cross norms (called thel-andm-norm, respectively) are introduced. These cross-norms (which depend on the order of factors, and are permuted when the latter is inverted) have the property that the respective completions ofE⊗F are Banach lattices under the ordering defined by the closure of the projective cone. Moreover, they are self-dual with respect to and coincide with well-known tensor norms in important special cases.