Novikov - Shubin signatures, I

Abstract

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.