Hilbertian versus Hilbert W*-modules, and Applications to L2- and Other Invariants

Hilbert(ian) A-modules over finite von Neumann algebras with a faithful normal trace state (from global analysis) and Hilbert W*-modules over A (from operator algebra theory) are compared, and a categorical equivalence is established. The correspondence between these two structures sheds new light on basic results in L-invariant theory providing alternative proofs. We indicate new invariants for finitely generated projective B-modules, where B is a unital C*-algebra, (usually the full group C*-algebra C∗(π) of the fundamental group π = π1(M) of a manifold M). During the last decade W. Lück, A. Carey, V. Mathai, and other authors [27, 32, 37, 6] introduced the analytical concept of Hilbert(ian) A-modules over finite von Neumann algebras A into global analysis. The method has been used to obtain L2-invariants for certain compact closed manifolds, e.g. L2-(co)homology, L2-torsion, L2-Betti numbers and Novikov-Shubin invariants for finitely generated Hilbert A-chain complexes. On the other hand Hilbert C*modules over arbitrary C*-algebras have been used in operator and operator algebra theory, in global analysis, in noncommutative geometry and mathematical physics for about 50 years, [23, 25, 47, 16]. The purpose of the present note is to compare these two categories of C -/C*valued inner product modules over finite von Neumann algebras A, where for technical purposes the von Neumann algebras A are supposed to admit a faithful normal trace state. We establish a categorical equivalence. Transferring known results on type II∞ von Neumann algebras and self-dual Hilbert W*-modules through this categorical equivalence to the theory of Hilbertian A-modules we obtain more evidence on the background of the theory of L2-invariants from a different viewpoint. We establish new invariants for finitely generated projective B-modules over unital C*-algebras B and give a perspective for future research.