  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 L^2 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 L^2 cohomology.
Under the determinant class assumption the L^2 torsions of this paper
specialize to the invariants studied in our previous work. Applying a recent
theorem of D. Burghelea, L. Friedlander and T. Kappeler we obtain a Cheeger -
Muller type theorem stating the equality between the combinatorial and the
analytic L^2 torsions.
