ar X iv : d g - ga / 9 61 00 02 v 1 3 O ct 1 99 6 DETERMINANT LINES , VON NEUMANN ALGEBRAS AND L 2 TORSION

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 combinatorial and L analytic torsion invariants associated to flat Hilbertian bundles over compact polyhedra and manifolds; we view them as volume forms on the reduced L homology and L 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 homology vanishes, the determinant line can be canonically identified with R, and our L torsion invariants specialize to the L 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 cohomology. §0. Introduction The study of the L Reidemeister-Franz torsion and the L analytic torsion was initiated in [M], [L] and [CM]. These invariants were originally defined for manifolds with trivial L cohomology and positive Novikov-Shubin invariants; they were shown to be piecewise linear and smooth invariants respectively. See also [LR] for K-theoretic generalizations of these invariants. In this paper, we introduce a new concept of determinant line of a finitely generated Hilbertian module over a von Neumann algebra. Here a Hilbertian module is defined as a topological vector space and a module over a von Neumann algebra such that there is an admissible scalar product on it, making it a Hilbert module. The construction of the determinant line, which we suggest here, generalizes the classical construction of determinant line of a finite dimensional vector space, and enjoys similar functorial properties. Nonzero elements of the determinant line can be naturally viewed as a volume forms on the Hilbertian module. This enables us to make sense of the notions of volume forms and determinant lines in the infinite dimensional and non-commutative situation. We then define analytic and the Reidemeister-Franz L torsion invariants of flat Hilbertian bundles of determinant class over finite polyhedra and compact manifolds respectively. These reduce to the classical constructions in the finite dimensional situation. These new torsion invariants live in the determinant lines of reduced L The research was supported by a grant from the US Israel Binational Science Foundation