Novikov - Shubin signatures, I

Torsion objects of von Neumann categories describe the phenomen “spectrum near zero” discovered by S. Novikov and M. Shubin. In this paper we classify Hermitian forms on torsion objects of a finite von Neumann category. We prove that any such form can be represented as a discriminant form of a degenerate Hermitian form on a projective module. We also find a relation between the Hermitian forms on projective modules which holds if and only if their discriminant forms are congruent. A notion of superfinite von Neumann category is introduced. It is proven that the classification of torsion Hermitian forms in a superfinite category can be completely reduced to the isomorphisn types of their positive and negative parts.