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
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.
2014 American Mathematical Society
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