L2 torsion without the determinant class condition and extended L2 cohomology

Abstract

We associate determinant lines to objects of the extended abelian category built out of a von Neumann category with a trace. Using this we suggest constructions of the combinatorial and the analytic L² torsions which, unlike the work of the previous authors, requires no additional assumptions; in particular we do not impose the determinant class condition. The resulting torsions are elements of the determinant line of the extended L² cohomology. Under the determinant class assumption the L² torsions of this paper specialize to the invariants studied in our previous work [6]. Applying a recent theorem of D. Burghelea, L. Friedlander and T. Kappeler [3] we obtain a Cheeger–Müller type theorem stating the equality between the combinatorial and the analytic L² torsions.