On tensor products of Banach lattices

Using techniques from harmonic analysis (more specifically Varopoulos' theory of tensor algebras) we study some tensor products of Banach lattices. Our main result is that if the Banach latticeB satisfies a certain additional property, the Gillespie factorization property, then a natural quotient of the spaceB^B is anL^l-space, and we identify the kernel of this quotient map.