Callias-type operators in von Neumann algebras

TL;DR

This paper extends index theory to Callias-type operators acting on sections of bundles over von Neumann algebras, proving index theorems and invariance properties, including a version of Atiyah's $L^2$-index theorem.

Contribution

It introduces new index theorems for Callias-type operators in von Neumann algebra settings and proves cobordism invariance of their indices.

Findings

Established a relative index theorem for von Neumann algebra operators.

Proved a Callias-type index theorem in the von Neumann algebra context.

Demonstrated cobordism invariance of the von Neumann index.

Abstract

We study differential operators on complete Riemannian manifolds which act on sections of a bundle of finite type modules over a von Neumann algebra with a trace. We prove a relative index and a Callias-type index theorems for von Neumann indexes of such operators. We apply these results to obtain a version of Atiyah's $L^2$-index theorem, which states that the index of a Callias-type operator on a non-compact manifold $M$ is equal to the $\Gamma$-index of its lift to a Galois cover of $M$. We also prove the cobordism invariance of the index of Callias-type operators. In particular, we give a new proof of the cobordism invariance of the von Neumann index of operators on compact manifolds.