De Rham Theorem for Extended L 2 -cohomology

We prove an analogue of the de Rham theorem for the extended L 2-cohomology introduced by M.Farber [Fa]. This is done by establishing that the de Rham complex over a compact closed manifold with coefficients in a flat Hilbert bundle E of A-modules over a finite von Neumann algebra A is chain-homotopy equivalent in the sense of [GS] (i.e. with bounded morphisms and homotopy operators) to a combinatorial complex with the same coefficients. This is established by using the Witten deformation of the de Rham complex. We also prove that the de Rham complex is chain-homotopy equivalent to the spectrally truncated de Rham complex which is also finitely generated.