CRASH COURSE ON TOPOLOGICAL K THEORY FOR C* ALGEBRAS

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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

Sujets

• Algebra
• Banach algebra
• Hilbert space
• multiplication

Informations

• Publié par: pefav
• Nombre de lectures: 13
• Langue: English

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CRASH-COURSEONTOPOLOGICALK-THEORYFOR C*-ALGEBRAS I.L.CHATTERJI Thistextisbasedonthefollowingsources,thatIrecommendtoany ofthosewhowanttolearnthesubjectalittlemoreseriously. [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“friendly approach”totopologicalK-theory,withalotofdetailsworked outcarefully.BasicsofC*-algebrasareassumed.) [6]A.Valette,IntroductiontotheBaum-Connesconjecture.(Chap- ter3givesashortintroductiontotopologicalK-theory.Basics ofC*-algebrasareassumed.) 1. Introduction Theaimofwhatfollowsistogiveacrash-courseinK-theoryofC*- algebras.Thistheorydefinesacollectionoffunctors { K n } n fromthe categoryofC*-algebrastothecategoryofabeliangroups,satisfyingthe Eilenberg-MacLaneaxiomsforahomologytheory 1 .Anicefeatureof thistheoryistheBottperiodicity,whichimpliesthatthosefunctorsare actuallyonlytwo.Thesubjectcoveredinthefollowingsectionsdoes workaswellforanyBanachalgebra.Thepointofconsideringonly C*-algebrasisfirstbecausemyfavouritemathematicalobjectshappen tobeC*-algebras 2 ,andsecondlybecausethisistheonlygenerality thatoneneedstohavetogetthelinkwiththeK-theoryoftopological Date:November7,2002. 1 Thissentenceisabstractnonsense,andunderstandingitisnotrequiredforthis text.ThesameappliestotherestoftheIntroduction. 2 ThereducedC*-algebraoftorsion-freegroups. 1