Index theory for locally compact noncommutative geometries

dc.contributor.authorCarey, Alan
dc.contributor.authorGayral, V.
dc.contributor.authorRennie, Adam
dc.contributor.authorSukochev, Fedor A
dc.date.accessioned2015-12-07T22:55:40Z
dc.date.available2015-12-07T22:55:40Z
dc.date.issued2014
dc.date.updated2016-06-14T09:13:25Z
dc.description.abstractSpectral 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. 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.en_AU
dc.format.extent130
dc.identifier.isbn9781470417215
dc.identifier.urihttp://hdl.handle.net/1885/28501
dc.publisherAmerican Mathematical Society
dc.relation.ispartofseriesMemoirs of the American Mathematical Society
dc.titleIndex theory for locally compact noncommutative geometries
dc.typeBook
dcterms.accessRightsOpen Access via publisher websiteen_AU
local.bibliographicCitation.placeofpublicationProvidence, USA
local.contributor.affiliationCarey, Alan, College of Physical and Mathematical Sciences, ANU
local.contributor.affiliationGayral, V., Universite Reims Champagne-Ardenne
local.contributor.affiliationRennie, Adam, University of Wollongong
local.contributor.affiliationSukochev, Fedor A, University of New South Wales
local.contributor.authoremailrepository.admin@anu.edu.au
local.contributor.authoruidCarey, Alan, u4043636
local.description.notesImported from ARIES
local.description.refereedYes
local.identifier.absfor010108 - Operator Algebras and Functional Analysis
local.identifier.absseo970101 - Expanding Knowledge in the Mathematical Sciences
local.identifier.ariespublicationu5328909xPUB58
local.identifier.doi10.1090/memo/1085
local.identifier.scopusID2-s2.0-84908140373
local.identifier.uidSubmittedByu5328909
local.type.statusMetadata onlyen_AU

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