Type II non-commutative geometry. I. Dixmier trace in von Neumann algebras

We define the notion of Connes–von Neumann spectral triple and consider the associated index problem. We compute the analytic Chern–Connes character of such a generalized spectral triple and prove the corresponding local formula for its Hochschild class. This formula involves the Dixmier trace for II∞ von Neumann algebras. In the case of foliations, we identify this Dixmier trace with the corresponding measured Wodzicki residue.