Abstract

Spectral triples for nonunital algebras model locally compact spaces in non-

commutative 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

L2-index

Theorem for covering spaces.

We also explain how to interpret the McKean-Singer formula in the nonunital case.

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.

Received by the editor July 5, 2011, and, in revised form, October 29, 2012.

Article electronically published on January 23, 2014.

DOI: http://dx.doi.org/10.1090/memo/1085

2010 Mathematics Subject Classification. Primary 46H30, 46L51, 46L80, 46L87, 19K35,

19K56, 58J05, 58J20, 58J30, 58J32, 58J42.

Key words and phrases. Local index formula, nonunital, spectral triple, Fredholm module,

Kasparov product.

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2014 American Mathematical Society

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