Reflexivity and the Grothendieck property for positive tensor products of Banach lattices-II

Abstract

Let ϕ be an Orlicz function such that ϕ and its complementary function ϕ* satisfy the Δ2-condition, let ℓϕ be an Orlicz sequence space associated to ϕ, and let X be a Banach lattice. Then ℓϕ ⊗F X (respectively, ℓϕ ∼⊗i X), the Fremlin projective (respectively, the Wittstock injective) tensor product of ℓϕ and X, has reflexivity or the Grothendieck property if and only if X has the same property and each positive linear operator from ℓϕ (respectively, from ℓϕ*) to X* (respectively, to X**) is compact. *The author is partially supported by the NSF of China, Grant No. 10871213. † The author is supported by the NSF of China, Grant No. 10671048.