ThisbookgivesanintroductiontoC*-algebrasandtheirrepresentationsonHilbertspaces.Wehavetriedtopresentonlywhatwebelievearethemostbasicideas,assimplyandconcretelyaswecould.Sowheneveritisconvenient(anditusuallyis),HilbertspacesbecomeseparableandC*-algebrasbecomeGCR.Thispracticeprobablycreatesanimpressionthatnothingofvalueisknownaboutother*-algebras.Ofcoursethatisnottrue.Butinsofarasrepresentationsarecon-cerned,wecanpointtotheempiricalfactthattothisdaynoonehasgivenaconcreteparametricdescriptionofeventheineduciblerepresentationsofany*-algebrawhichisnotGCR.Indeed,thereismetamathematicalevidencewhichstronglysuggeststhatnooneeverwill(seethediscussionattheendofSection3.4).Occasionally,whentheideabehindtheproofofageneraltheoremisexposedveryclearlyinaspecialcase,weproveonlythespecialcaseandrelegategeneralizationstotheexercises.Ineffect,wehavesystematicallyeschewedtheBourbakitradition.Wehavealsotriedtotakeintoaccounttheinterestsofavarietyofreaders.Forexample,themultiplicitytheoryfornormaloperatorsiscontainedinSections2.Iand2.2.(itwouldbedesirablebutnotnecessarytoincludeSectionI.1aswell),whereassomeoneinterestedinBoreLstructurescouldread