  In this paper, we suggest a construction of determinant lines of finitely
generated Hilbertian modules over finite von Neumann algebras. Nonzero elements
of the determinant lines can be viewed as volume forms on the Hilbertian
modules. Using this, we study both $L^2$ combinatorial and $L^2$ analytic
torsion invariants associated to flat Hilbertian bundles over compact polyhedra
and manifolds; we view them as volume forms on the reduced $L^2$ homology and
cohomology. These torsion invariants specialize to the the classical
Reidemeister-Franz torsion and the Ray-Singer torsion in the finite dimensional
case. Under the assumption that the $L^2$ homology vanishes, the determinant
line can be canonically identified with $\R$, and our $L^2$ torsion invariants
specialize to the $L^2$ torsion invariants previously constructed by A.Carey,
V.Mathai and J.Lott. We also show that a recent theorem of Burghelea et al. can
be reformulated as stating equality between two volume forms (the combinatorial
and the analytic) on the reduced $L^2$ cohomology.
