notes on operator algebras

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1、NotesonOperatorAlgebrasG.JungmanSchoolofNaturalSciences,InstituteforAdvancedStudyOldenLane,Princeton,NJ08540Contents1StructureTheoryI41.1InvertibleElementsandSpectra............................41.2GelfandTransform...................................71.3LocalAlgebras,IdempotentsandProjections...

2、..................92vonNeumannAlgebras112.1CommutantandBicommutant.............................112.2Factors..........................................122.3TheTrace........................................122.4HyperfiniteAlgebras..................................133StatesandRepresentations143.1GNS

3、Construction....................................143.2BasicStructureofRepresentations...........................164StructureTheoryII214.1WeightsandTraces...................................214.2Types..........................................235Matrices275.1InductiveLimits........................

4、.............275.2GlimmAlgebras.....................................275.3Matn(A)........................................306AutomorphismGroups316.1AutomorphismsandInvariantStates..........................316.2KMSStates.......................................396.3ModularGroup.....................

5、................426.4TypeIIIFactors....................................472CONTENTS36.5HyperfinitenessAgain.................................487Extensions498K-Theory508.1Introduction.......................................508.2CommutativeK-Theory................................508.3K0-Theory......

6、..................................538.4K1-Theory........................................558.5AFAlgebras.......................................568.6EquivariantK-Theory.................................568.7IndexTheorems.....................................579NuclearCAlgebras59Chapter1StructureT

7、heoryI1.1InvertibleElementsandSpectraDefinition.ACalgebraisaninvolutiveBanachalgebraAwithanormsatisfyingtherelationskABkkAkkBk;kAk=kAk;2kAAk=kAk:WedenotethedualalgebrabyA0.ThenwecanandoftenwillusetheweaktopologyforAandtheweak-*topolo