Hilbertian versus Hilbert W*-modules, and applications to $L^2$- and other invariants
Michael Frank
TL;DR
The paper establishes a categorical equivalence between Hilbertian $A$-modules over a finite von Neumann algebra $A$ with a trace and self-dual Hilbert $A$-W*-modules, via a dilation functor that yields an embedding-independent canonical object. This equivalence aligns finitely generated Hilbertian modules with finitely generated projective $A$-modules and reframes End$_A$-algebras as corners of type ${\rm II}_\infty$ algebras, enabling transfer of results from type ${\rm II}_\infty$ theory to the Hilbertian module setting. The work introduces center-valued trace criteria for finite generation and leverages modular frames to classify finitely generated Hilbert modules over unital C*-algebras, providing new invariants relevant to $L^2$-invariant theory. It also discusses implications for using finitely generated projective $C^*(\pi)$-modules in place of von Neumann completions in $L^2$-invariants and outlines directions for future investigations into transfer matrices and 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 $C^*(π)$ of the fundamental group $π=π_1(M)$ of a manifold $M$). The results are of interest to specialists in operator algebras and global analysis.