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CRASH COURSE ON TOPOLOGICAL K THEORY FOR C* ALGEBRAS

13 pages
CRASH-COURSE ON TOPOLOGICAL K-THEORY FOR C*-ALGEBRAS I. L. CHATTERJI This text is based on the following sources, that I recommend to any of those who want to learn the subject a little more seriously. [1] J.B. Conway, A Course in Functional Analysis. (Basics con- cerning C?-algebras.) [2] G. J. Murphy, C?-algebras and operator theory. (Basics con- cerning C?-algebras, with some K-theory.) [3] M. Takesaki, Theory of Operators algebras I. (Basics concern- ing C?-algebras.) [4] J.L. Taylor, Banach Algebras and Topology. (A nice introduc- tion to topological K-theory.) [5] N. E. Wegge-Olsen, K-Theory and C*-algebras. (A “friendly approach” to topological K-theory, with a lot of details worked out carefully. Basics of C*-algebras are assumed.) [6] A. Valette, Introduction to the Baum-Connes conjecture. (Chap- ter 3 gives a short introduction to topological K-theory. Basics of C*-algebras are assumed.) 1. Introduction The aim of what follows is to give a crash-course in K-theory of C*- algebras.

  • algebras

  • group

  • algebras determines

  • report any

  • banach algebra

  • any infinite

  • hilbert space

  • compact hausdorff

  • any such operator


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CRASH-COURSEONTOPOLOGICALK-THEORYFORC*-ALGEBRASI.L.CHATTERJIThistextisbasedonthefollowingsources,thatIrecommendtoanyofthosewhowanttolearnthesubjectalittlemoreseriously.[1]J.B.Conway,ACourseinFunctionalAnalysis.(Basicscon-cerningC-algebras.)[2]G.J.Murphy,C-algebrasandoperatortheory.(Basicscon-cerningC-algebras,withsomeK-theory.)[3]M.Takesaki,TheoryofOperatorsalgebrasI.(Basicsconcern-ingC-algebras.)[4]J.L.Taylor,BanachAlgebrasandTopology.(Aniceintroduc-tiontotopologicalK-theory.)[5]N.E.Wegge-Olsen,K-TheoryandC*-algebras.(A“friendlyapproach”totopologicalK-theory,withalotofdetailsworkedoutcarefully.BasicsofC*-algebrasareassumed.)[6]A.Valette,IntroductiontotheBaum-Connesconjecture.(Chap-ter3givesashortintroductiontotopologicalK-theory.BasicsofC*-algebrasareassumed.)1.IntroductionTheaimofwhatfollowsistogiveacrash-courseinK-theoryofC*-algebras.Thistheorydefinesacollectionoffunctors{Kn}nfromthecategoryofC*-algebrastothecategoryofabeliangroups,satisfyingtheEilenberg-MacLaneaxiomsforahomologytheory1.AnicefeatureofthistheoryistheBottperiodicity,whichimpliesthatthosefunctorsareactuallyonlytwo.ThesubjectcoveredinthefollowingsectionsdoesworkaswellforanyBanachalgebra.ThepointofconsideringonlyC*-algebrasisfirstbecausemyfavouritemathematicalobjectshappentobeC*-algebras2,andsecondlybecausethisistheonlygeneralitythatoneneedstohavetogetthelinkwiththeK-theoryoftopologicalDate:November7,2002.1Thissentenceisabstractnonsense,andunderstandingitisnotrequiredforthistext.ThesameappliestotherestoftheIntroduction.2ThereducedC*-algebraoftorsion-freegroups.1