$L^{2}$-von Neumann modules, their relative tensor products and the spatial derivative

Abstract

We develop a theory of $L^{2}$-von Neumann modules, which encompasses a reformulation of Connes' Spatial Derivative, and the Relative Tensor Product of Sauvageot. We demonstrate the naturality of the relative tensor product construction in the category of $L^{2}$-von Neumann bimodules. Finally, we give evidence for the claim that the relative tensor product is essentially the only tensor product which should be used when considering this tensor category.