Index theory for locally compact noncommutative geometries

A. L. Carey, V. Gayral, A. Rennie, F. A. Sukochev

    Research output: Contribution to journalArticlepeer-review

    55 Citations (Scopus)

    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 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.

    Original languageEnglish
    Pages (from-to)1-142
    Number of pages142
    JournalMemoirs of the American Mathematical Society
    Volume231
    Issue number1085
    DOIs
    Publication statusPublished - 1 Sept 2014

    Fingerprint

    Dive into the research topics of 'Index theory for locally compact noncommutative geometries'. Together they form a unique fingerprint.

    Cite this