We prove that the multiplication map L a (M ) ⊗ M L b (M ) → 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 ⊗ 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 ) → 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 ≥ 0.