资源描述：
《非交换微分几何（孔涅）》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、NON-COMMUTATIVEDIFFERENTIALGEOMETRYbyALAINCO~N-~q'ESIntroductionThisistheintroductiontoaseriesofpapersinwhichweshallextendthecalculusofdifferentialformsandthedeRhamhomologyofcurrentsbeyondtheircustomaryframeworkofmanifolds,inordertodealwithspacesofamoreelaboratenature,suchas,a)thespaceofleavesofaf
2、oliation,b)thedualspaceofafinitelygeneratednon-abeliandiscretegroup(orLiegroup),c)theorbitspaceoftheactionofadiscretegroup(orLiegroup)onamanifold.Whatsuchspaceshaveincommonistobe,ingeneral,badlybehavedaspointsets,sothattheusualtoolsofmeasuretheory,topologyanddifferentialgeometrylosetheirpertinence
3、.Thesespacesaremuchbetterunderstoodbymeansofacanonicallyassociatedalgebrawhichisthegroupconvolutionalgebraincaseb).WhenthespaceVisanordinarymanifold,theassociatedalgebraiscommutative.Itisanalgebraofcomplex-valuedfunctionsonV,endowedwiththepointwiseoperationsofsumandproduct.AsmoothmanifoldVcanbecon
4、sideredfromdifferentpointsofviewsuchaso~)Measuretheory(i.e.Vappearsasameasurespacewithafixedmeasureclass),8)Topology(i.e.Vappearsasalocallycompactspace),y)Differentialgeometry(i.e.Vappearsasasmoothmanifold).EachofthesestructuresonVisfullyspecifiedbythecorrespondingalgebraoffunctions,namely:~)Theco
5、mmutativevonNeumannalgebraL~ofclassesofessentiallyboundedmeasurablefunctionsonV,8)The&-algebraG0(V)ofcontinuousfunctionsonVwhichvanishatinfinity,y)ThealgebraG~~ofsmoothfunctionswithcompactsupport.257642ALAINCONNESIthaslongbeenknowntooperatoralgebraiststhatmeasuretheoryandtopologyextendfarbeyondthe
6、irusualframeworktoA)ThetheoryofweightsandyonaVeumannalgebras,B)C*-algebras,K-theoryandindextheory.Letusbrieflydiscussthesetwofields,A)ThetheoryofweightsandyonNeumannalgebrasToanordinarymeasurespace(X,$)correspondthevonNeumannalgebraL~~~)andtheweight~:r=fxfdVfeL~X,?t)+"Anypair(M,~)ofacommutativevon
7、NeumannalgebraMandweightq~isobtainedinthiswayfromameasurespace(X,~).ThustheplaceofordinarymeasuretheoryinthetheoryofweightsonvonNeumannalgebrasissimilartothatofcommutativealgebrasamongarbitraryones.ThisiswhyA)iso
