Determinant lines, von Neumann algebras and $L^2$ 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 $L^2$ 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 ReidemeisterFranz torsion and the RaySinger 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.
 Publication:

eprint arXiv:dgga/961000
 Pub Date:
 October 1996
 arXiv:
 arXiv:dgga/9610002
 Bibcode:
 1996dg.ga....10002C
 Keywords:

 Mathematics  Differential Geometry
 EPrint:
 AMSTex, 27 pages