Some geometric properties inherited by the positive tensor products of atomic Banach lattices

For Banach lattices XX and YY, let X⊗ˆ|π|Y and X⊗̌|ε|Y denote the positive projective and injective tensor products of XX and YY, respectively. In this paper, we characterize the inheritance of reflexivity and containment of copies of c0,ℓ1,ℓ∞c0,ℓ1,ℓ∞ by X⊗ˆ|π|Y and X⊗̌|ε|Y from XX and YY, when one of them is an atomic Banach lattice. In this case, we also give an affirmative answer to an open question of Jeurnink.