The Radon–Nikodym Property for Tensor Products of Banach Lattices

Abstract

Let 1 p < ∞. We show that p⊗F X, the Fremlin projective 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 that p⊗i X, 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 from p to X is compact, where

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

“…However, essentially new ideas come in via the Matuszewska-Orlicz indices and Schauder decompositions. Thus the results in this paper are a wide generalization of those in [2].…”

“…Then X⊗ i Y with C i as its positive cone is a Banach lattice (see [14,15] or [11, §3.8]), called the Wittstock injective tensor product of X and Y . Identical to the proof of Theorem 11 in [2], we have…”

“…We leave the proof of the following proposition, almost identical to the proofs of Theorem 8 and Theorem 9 in [2], to the readers. Proposition 3.4.…”

“…In [2], Bu and Buskes discussed the inheritance of the Radon-Nikodym property from a Banach lattice X to the Fremlin projective tensor product p⊗F X and the Wittstock injective tensor product p⊗i X for 1 ≤ p < ∞. In this paper, building on the idea of [2], we will first give sequential representations of the Fremlin projective tensor product ϕ⊗F X and the Wittstock injective tensor product ϕ⊗i X of an Orlicz sequence space ϕ with a Banach lattice X. Then we use these sequential representations to characterize the inheritance of the Radon-Nikodym property from an Orlicz sequence space ϕ and a Banach lattice X to their Fremlin projective tensor product and Wittstock injective tensor product.…”

“…Then we use these sequential representations to characterize the inheritance of the Radon-Nikodym property from an Orlicz sequence space ϕ and a Banach lattice X to their Fremlin projective tensor product and Wittstock injective tensor product. We note that some of the details in this paper are virtually identical to [2] and where they are we have not included the identical details. However, essentially new ideas come in via the Matuszewska-Orlicz indices and Schauder decompositions.…”