projective spectrum in banach algebras

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1、PROJECTIVESPECTRUMINBANACHALGEBRASRONGWEIYANG*Abstract.ForatupleA=(A0,A1,...,An)ofelementsinaunitalBanachalgebraB,itsprojectivespectrump(A)isdefinedtobethecollectionofz=[z0,z1,...,zn]∈PnsuchthatA(z)=z0A0+z1A1+···+znAnisnotinvertibleinB.Thepre-imageofp(A)inCn+1isdenotedbyP(A).WhenBisthek×

2、kmatrixalgebraMk(C),theprojectivespectrumisaprojectivehypersurface.Ininfinitedimensionalcases,projectivespectrumscanbeverycomplicated,butalsohavesomepropertiessimilartothatofhypersurfaces.WhenAiscommutative,P(A)isaunionofhyperplanes.WhenBisreflexiveorisaC∗-algebra,theprojectiveresolventse

3、tPc(A):=Cn+1P(A)isshowntobeadisjointunionofdomainsofholomorphy.LaterpartofthispaperstudiesMaurer-CartantypeB-valued1-formA−1(z)dA(z)onPc(A).Asaconsequence,weshowthatifBisaC∗-algebrawithatraceφ,thenφ(A−1(z)dA(z))isanontrivialelementinthedeRhamcohomologyspaceH1(Pc(A),C).d0.IntroductionTh

4、eclassicalspectrumofanelementAinaunitalBanachalgebraBisdefinedthrougharXiv:0804.0387v1[math.FA]2Apr2008theinvertibilityofA−λI.IfA=(A0,A1,...,An)isacommutativetupleofele-mentsinB,thenclassicalnotionsofjointspectrumaredefinedthroughtheinvertibilityof(A0−λ0I,A1−λ1I,...,An−λnI)invarioussenses

5、(H¨ormander[H¨o]Ch3,andTaylor[Ta]).Inallthesecases,theidentityIservesasabaseagainstwhichtheinvertibilitiesofotherelementsaremeasured.Theideaofprojectivespectrum,whichwewilldefineandstudy,istosetIfree,andconsidertheinvertibilityofz0A0+z1A1,ormoregenerally,A(z):=z0A0+z1A1+···+znAn.Thisisam

6、easurementofhowtheelementsbehaveagainst1991MathematicsSubjectClassification.Primary47A13;Secondary47L10.Keywordsandphrases.Banachalgebra,centrallinearfunctional,Maurer-Cartanform,maximalidealspace,projectivespectrum,projectiveresolventset,deRhamcohomology,unionofhyperplanes.Thisworkissup

7、portedinpartbyagrantfromtheNationalScienceFoundation(DMS0500333).12R.YANGeachother.Unlikeclassicalnotionsofjointspectrums,projectivespectrumisdefinedforalltuples,notjustcommutativeones.Thispaperisorganizedasfollows.Section1.Preparation.Herewedefinetheprojectivespectrumandproveits