  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.
