Copies of ℓ 1 in positive tensor products of Orlicz sequence spaces

Abstract

Let X be a Banach lattice and ℓ ϕ be an Orlicz sequence space associated to an Orlicz function ϕ with the Δ2-condition. In this paper, we prove that (i) the positive injective tensor product of ℓ ϕ and X, contains no copy of ℓ1 if and only if both ℓ ϕ and X contain no copy of ℓ 1; and (ii) the positive projective tensor product of ℓ ϕ and X, contains no copy of ℓ 1 if and only if both ℓ ϕ and X contain no copy of ℓ 1 and each positive linear operator from ℓ ϕ to X* is compact.