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1、PUBLICATIONSMATHÉMATIQUESDEL’I.H.É.S.ALAINCONNESNon-commutativedifferentialgeometryPublicationsmathématiquesdel’I.H.É.S.,tome62,no2(1985),p.41-144.©Publicationsmathématiquesdel’I.H.É.S.,1985,tousdroitsréservés.L’
2、accèsauxarchivesdelarevue«Publicationsmathématiquesdel’I.H.É.S.»(http://www.ihes.fr/IHES/Publications/Publications.html),impliquel’accordaveclesconditionsgénéralesd’utilisation(http://www.numdam.org/legal.php).Touteutilisationcommercialeouimpressionsystématiqueestc
3、onstitutived’uneinfractionpénale.Toutecopieouimpressiondecefi-chierdoitcontenirlaprésentementiondecopyright.ArticlenumérisédanslecadreduprogrammeNumérisationdedocumentsanciensmathématiqueshttp://www.numdam.org/NON-COMMUTATIVEDIFFERENTIALGEOMETRYbyALAINGONNESIntroduc
4、tionThisistheintroductiontoaseriesofpapersinwhichweshallextendthecalculusofdifferentialformsandthedeRhamhomologyofcurrentsbeyondtheircustomaryframeworkofmanifolds,inordertodealwithspacesofamoreelaboratenature,suchas,a)thespaceofleavesofafoliation,b)thedualspaceofaf
5、initelygeneratednon-abeliandiscretegroup(orLiegroup),c)theorbitspaceoftheactionofadiscretegroup(orLiegroup)onamanifold.Whatsuchspaceshaveincommonistobe,ingeneral,badlybehavedaspointsets,sothattheusualtoolsofmeasuretheory,topologyanddifferentialgeometrylosetheirpert
6、inence.Thesespacesaremuchbetterunderstoodbymeansofacanonicallyassociatedalgebrawhichisthegroupconvolutionalgebraincaseb).WhenthespaceVisanordinarymanifold,theassociatedalgebraiscommutative.Itisanalgebraofcomplex-valuedfunctionsonV,endowedwiththepointwiseoperationso
7、fsumandproduct.AsmoothmanifoldVcanbeconsideredfromdifferentpointsofviewsuchasa)Measuretheory(i.e.Vappearsasameasurespacewithafixedmeasureclass),P)Topology(i.e.Vappearsasalocallycompactspace),y)Differentialgeometry(i.e.Vappearsasasmoothmanifold).Eachofthesestructure
8、sonVisfullyspecifiedbythecorrespondingalgebraoffunctions,namely:a)ThecommutativevonNeumannalgebraL°°(V)ofclassesofessentiallyboundedmeasurablefun
