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

Abstract

Let X be a Banach lattice and p, p′ be real numbers such that 1 < p, p′<∞ and 1/p + 1/p′ = 1. Then $${\ell_p\hat{\otimes}_FX}$$ (respectively, $${\ell_p\tilde{\otimes}_{i}X}$$ ), the Fremlin projective (respectively, the Wittstock injective) tensor product of l p and X, has reflexivity or the Grothendieck property if and only if X has the same property and each positive linear operator from l p (respectively, from l p′) to X* (respectively, to X**) is compact.