; TeX output 2001.10.12:2359 e̍6XIhtml: html:f cmcsc8DocumentuUa#Math. :K`y
cmr10241+46XIehtml: html: A_-
cmcsc10ExtensionsofSt32able
b>
cmmi10C ^O! cmsy7P -Algebras#H؍ ŷ-
cmcsc10MikaelRrdam{L|{Y cmr8ReceivÎed: AugustX8,2001lkCommÎunicatedXbyJoachimCuntz) Abstract.W*expshowthatanextensionoftwostableC ^P -algebras needUUnotbGestable.qMoreexplicitlywendanextensionic0
!",
cmsy10!C (Z)8
K!A!K!0 for[some(innitedimensional)compactHausdorspaceZѲsuchthat Aisnotstable.<>TheC ^P -algebraAinourexamplehasanapproximate unitUUconsistingofpro 8jections.l 2000UUMathematicsSub 8jectClassication:q46L05,46L35 KeywordsUUandPhrases:qstableC ^P -algebras,extensions@html: html:
91KIntroductionIt'followsfromBDF-theory*,[#html:1 html: ],that'foreveryextension0!K!A!BG!0ofХseparableC ^P -algebrasonehasAisstableifandonlyifBQisstable.EThisfactpromptedMthequestionifeveryextensionoftwo(separable)stableC ^P -algebrasisUUstable,aquestionwehereanswerinthenegative.InanearlierpapGerwithJ.Hjelmborg,[#html:3 html: ],wederivedacharacterizationofstabilityforC ^P -algebras,actuallyinthehopGeofprovidingapGositiveanswertothesjextensionproblem.Later,zin[html:5 html: ],theauthorshowedsjthatstabilityisnotanicely^bGehavedpropGertybyprovidinganexampleofa(simple,a3separable)C ^P -algebra/AsuchthatMٓR cmr72|s(A)isstablewhileAisnon-stable.JTheconstructionofthatN*examplewasinspiredbyideasofVilladsenfrom[%html:9 html: ].ocAgainusingideasofVilladsenandofresultsobtainedin[html:5 html: ]and[html:6 html:]theauthorfoundin[ html:8 html:]anexam-pledofasimpleC ^P -algebrathatcontainsbGothanon-zeroniteandaninnitepro 8jection.mAG?keyGBingredientinthisconstructionwasastudyofpro 8jectionsinthemultiplieralgebraofC (Z)Y
K$,whereZ=istheinniteCartesianproGductofL2-spheres.^Inparticular,&arecipGewasderivedfordecidingwhencertainpro-jectionsminthismultiplieralgebra,sarisingasinnitesumsofBottpro 8jections,are
propGerlyinnite.Thisrecipe(restatedhereinProposition2.1)isalsoacrucialUUingredientintheconstructionoftheexamplegiveninthisnote. DocumentuUa#Mathematica6(2001)241{246 *e̍6XIhtml: html:242 ŷMikaelRrdam+46XIehtml: html:
2KTheConstructionW*etreviewsomeofthenotationandsomeoftheresultsfrom[ html:8 html: ].%Let Z'Xdenotep