Equivalence of the Bochner norm and its transpose characterizes Lp-spaces

We show that a Banach lattice E is Riesz and topologically (isometrically) isomorphic to an Lp(μ)-space, for some measure space (Ω,Σ,μ), if and only if the Bochner norm Δp is equivalent (equal) to its transpose Δpt on Lp(μ1)⊗E, where the measure space (Ω1,Σ1,μ1) may be taken as any σ-finite measure space or any probability space. The characterizations presented uses properties of the sequence space E(ℓp) of Krivine.