Hilbertian Versus Hilbert W*-Modules and Applications to L2- and other invariants

Abstract

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 2 -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 ∗ (�) of the fundamental group � = �1(M) of a manifold M).