Index theory for locally compact noncommutative geometries

TL;DR

This paper extends the local index formula to spectral triples over nonunital algebras in noncommutative geometry, providing new tools for index calculations in genuinely noncommutative and nonunital settings, with applications to manifolds of bounded geometry.

Contribution

It proves the local index formula for nonunital spectral triples without local units, including new results for odd-dimensional manifolds and a version of Atiyah's L^2-index Theorem.

Findings

Established local index formula for nonunital algebras

Derived an analogue of Gromov-Lawson index formula for bounded geometry

Proved a version of Atiyah's L^2-index Theorem for covering spaces

Abstract

Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, we prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra. This formula has been successfully used to calculate index pairings in numerous noncommutative examples. The absence of any other effective method of investigating index problems in geometries that are genuinely noncommutative, particularly in the nonunital situation, was a primary motivation for this study and we illustrate this point with two examples in the text. In order to understand what is new in our approach in the commutative setting we prove an analogue of the Gromov-Lawson relative index formula (for Dirac type operators) for even dimensional manifolds with bounded geometry, without invoking compact supports. For odd dimensional manifolds our index formula appears to be completely new. As we prove our local index formula in the framework of semifinite noncommutative geometry we are also able to prove, for manifolds of bounded geometry, a version of Atiyah's L^2-index Theorem for covering spaces. We also explain how to interpret the McKean-Singer formula in the nonunital case. In order to prove the local index formula, we develop an integration theory compatible with a refinement of the existing pseudodifferential calculus for spectral triples. We also clarify some aspects of index theory for nonunital algebras.