Determinant lines, von Neumann algebras and L2 torsion

Abstract

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 L2 combinatorial and L2 analytic torsion invariants associated to flat Hilbertian bundles over compact polyhedra and manifolds; we view them as volume forms on the reduced L2 homology and L2 cohomology. These torsion invariants specialize to the classical Reidemeister-Franz torsion and the Ray-Singer torsion in the finite dimensional case. Under the assumption that the L2 homology vanishes, the determinant line can be canonically identified with ℝ, and our L2 torsion invariants specialize to the L2 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 L2 cohomology. Show more