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Determinant lines, von Neumann algebras and \(L^ 2\) torsion. (English) Zbl 0872.46031
We suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of these determinant lines can be viewed as volume forms on the Hilbertian modules.
Using this construction, 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 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 \(\mathbb{R}\) 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, Friedlander, Kappeler and McDonald can be reformulated as stating equality between two volume forms (the combinatorial and the analytic) on the reduced \(L^2\) cohomology.
Reviewer: A.Carey (Adelaide)

MSC:
46L10 General theory of von Neumann algebras
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
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