Spectral theory of von Neumann algebra valued differential operators over non-compact manifolds

TL;DR

This paper develops spectral theory for differential operators on infinite-dimensional bundles over non-compact manifolds, extending index theory and stability results to von Neumann algebra-valued contexts.

Contribution

It introduces criteria for self-adjointness and Fredholmness of differential operators on von Neumann algebra bundles and extends Callias-type index theory to this setting.

Findings

Criteria for self-adjointness of operators

Extension of Callias index to von Neumann algebra bundles

Index stability under compact perturbations

Abstract

We provide criteria for self-adjointness and {\tau}-Fredhomness of first and second order differential operators acting on sections of infinite dimensional bundles, whose fibers are modules of finite type over a von Neumann algebra A endowed with a trace {\tau}. We extend the Callias-type index to operators acting on sections of such bundles and show that this index is stable under compact perturbations.