Callias-type operators in von Neumann algebras

2016/02/22 by Braverman, Maxim, Cecchini, Simone
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1602.06873

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 L²-index theorem, which states that the index of a Callias-type operator on a non-compact manifold M is equal to the Γ-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.

Related

• Cobordism Invariance of the Index of Callias-Type Operators
• Spectral theory of von Neumann algebra valued differential operators\n over non-compact manifolds
• The index of Callias-type operators with Atiyah-Patodi-Singer boundary conditions
• Index theory in spaces of manifolds
• Callias-type operators in C^∗-algebras and positive scalar curvature on noncompact manifolds