Analytic and Reidemeister torsion for representations in finite type Hilbert modules

For a closed Riemannian manifold (M, g) we extend the definition of analytic and Reidemeister torsion associated to a unitary representation of π₁(M) on a finite dimensional vector space to a representation on a A-Hilbert module W of finite type where A is a finite von Neumann algebra. If (M, W) is of determinant class we prove, generalizing the Cheeger-Müller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, the L₂-analytic and L₂-Reidemeister torsions are equal.