Hilbertian versus Hilbert W*-modules, and applications to $L^2$- and other invariants

Abstract

Hilbert(ian) A-modules over finite von Neumann algebras A 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 supposed any unital C*-algebra, (usually the full group C*-algebra $C^{\ast }(\pi )$ of the fundamental group $\pi ={\pi }_1(M)$ of a manifold $M$). The results are of interest to specialists in operator algebras and global analysis.

12 pages, Latex2e

References in corpus (2)

• Frames in Hilbert C*-modules and C*-algebras
• Novikov-Shubin invariants for arbitrary group actions and their positivity

Cited by in corpus (1)

• Amenability and vanishing of L^2-Betti numbers: an operator algebraic approach