Algebraic tensor products and internal homs of noncommutative L_p-spaces

We prove that the multiplication map L_a(M)\otimes_M L_b(M)\to L_{a+b}(M) is an isometric isomorphism of (quasi)Banach M-M-bimodules. Here L_a(M)=L^{1/a}(M) is the noncommutative L_p-space of an arbitrary von Neumann algebra M and \otimes_M denotes the algebraic tensor product over M equipped with the (quasi)projective tensor norm, but without any kind of completion. Similarly, the left multiplication map L_a(M)\to Hom_M(L_b(M),L_{a+b}(M)) is an isometric isomorphism of (quasi)Banach M-M-bimodules, where Hom_M denotes the algebraic internal hom without any continuity assumptions. In a forthcoming paper these results will be applied to L_p-modules introduced by Junge and Sherman, establishing explicit algebraic equivalences between the categories of right L_p(M)-modules for all p\ge0.