Novikov-Shubin signatures. I (Q1590084)

The author studies Hermitian forms on torsion objects of a finite von Neumann category. These subcategories of the category of Hilbertian spaces and continuous linear maps arise, e.g., from Hilbert representations of a discrete group. The general theory of Hermitian forms in additive categories was developed by \textit{H.-G. Quebbemann, W. Scharlau} and \textit{M. Schulte} [``Quadratic and Hermitian forms in additive and abelian categories'', J. Algebra 59, 264-289; (1979; Zbl 0412.18016)]. The main result shows that any non-degenerate Hermitian form on a torsion object can be represented as the discriminant of a degenerate form on a projective module. Furthermore, it is shown that any such Hermitian form is the orthogonal sum of a positively and of a negatively definite form, and that under some further assumptions (superfiniteness) this decomposition is unique.