Non Associative Normed Algebras
Volume 2 Representation Theory and the Zel manov Approach 1st Edition
Miguel Cabrera García
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NON-ASSOCIATIVENORMEDALGEBRAS
Volume2:RepresentationTheoryandtheZel’manovApproach
Thisfirstsystematicaccountofthebasictheoryofnormedalgebras,without assumingassociativity,includesmanynewandunpublishedresultsandissureto becomeacentralresourceforresearchersandgraduatestudentsinthefield.
Thissecondvolumerevisits JB*-triples,coversZel’manov’scelebratedworkin Jordantheory,provestheunit-freevariantofVidav–Palmertheorem,anddevelops therepresentationtheoryofalternative C *-algebrasandnon-commutative JB*-algebras.Thiscompletestheworkbeguninthefirstvolume,whichintroduced thesealgebrasanddiscussedtheso-callednon-associativeGelfand–Naimarkand Vidav–Palmertheorems.
Thisbookinterweavespurealgebra,geometryofnormedspaces,and infinite-dimensionalcomplexanalysis.Novelproofsarepresentedincomplete detailatalevelaccessibletograduatestudents.Thebookcontainsawealthof historicalcomments,backgroundmaterial,examples,andanextensivebibliography.
EncyclopediaofMathematicsandItsApplications
Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplicationin mathematicsormathematicalscienceandforwhichadetaileddevelopmentofthe abstracttheoryislessimportantthanathoroughandconcreteexplorationofthe implicationsandapplications.
Booksinthe EncyclopediaofMathematicsandItsApplications covertheir subjectscomprehensively.Lessimportantresultsmaybesummarizedasexercises attheendsofchapters.Fortechnicalities,readerscanbereferredtothe bibliography,whichisexpectedtobecomprehensive.Asaresult,volumesare encyclopedicreferencesormanageableguidestomajorsubjects.
EncyclopediaofMathematicsanditsApplications
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress.Foracompleteserieslistingvisit www.cambridge.org/mathematics.
119M.DezaandM.DutourSikiri ´ c GeometryofChemicalGraphs
120T.Nishiura AbsoluteMeasurableSpaces
121M.Prest Purity,SpectraandLocalisation
122S.Khrushchev OrthogonalPolynomialsandContinuedFractions
123H.NagamochiandT.Ibaraki AlgorithmicAspectsofGraphConnectivity
124F.W.King HilbertTransformsI
125F.W.King HilbertTransformsII
126O.CalinandD.-C.Chang Sub-RiemannianGeometry
127M.Grabisch etal.AggregationFunctions
128L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.Tucker TopicsinTopological GraphTheory
129J.Berstel,D.PerrinandC.Reutenauer CodesandAutomata
130T.G.Faticoni ModulesoverEndomorphismRings
131H.Morimoto StochasticControlandMathematicalModeling
132G.Schmidt RelationalMathematics
133P.KornerupandD.W.Matula FinitePrecisionNumberSystemsandArithmetic
134Y.CramaandP.L.Hammer(eds.) BooleanModelsandMethodsinMathematics,Computer Science,andEngineering
135V.Berth ´ eandM.Rigo(eds.) Combinatorics,AutomataandNumberTheory
136A.Krist ´ aly,V.D.R ˘ adulescuandC.Varga VariationalPrinciplesinMathematicalPhysics, Geometry,andEconomics
137J.BerstelandC.Reutenauer NoncommutativeRationalSerieswithApplications
138B.CourcelleandJ.Engelfriet GraphStructureandMonadicSecond-OrderLogic
139M.Fiedler MatricesandGraphsinGeometry
140N.Vakil RealAnalysisthroughModernInfinitesimals
141R.B.Paris HadamardExpansionsandHyperasymptoticEvaluation
142Y.CramaandP.L.Hammer BooleanFunctions
143A.Arapostathis,V.S.Borkar,andM.K.Ghosh ErgodicControlofDiffusionProcesses
144N.Caspard,B.Leclerc,andB.Monjardet FiniteOrderedSets
145D.Z.ArovandH.Dym BitangentialDirectandInverseProblemsforSystemsofIntegraland DifferentialEquations
146G.Dassios EllipsoidalHarmonics
147L.W.BeinekeandR.J.Wilson(eds.)withO.R.Oellermann TopicsinStructuralGraphTheory
148L.Berlyand,A.G.Kolpakov,andA.Novikov IntroductiontotheNetworkApproximationMethod forMaterialsModeling
149M.BaakeandU.Grimm AperiodicOrderI:AMathematicalInvitation
150J.Borwein etal.LatticeSumsThenandNow
151R.Schneider ConvexBodies:TheBrunn–MinkowskiTheory(SecondEdition)
152G.DaPratoandJ.Zabczyk StochasticEquationsinInfiniteDimensions(SecondEdition)
153D.Hofmann,G.J.Seal,andW.Tholen(eds.) MonoidalTopology
154M.CabreraGarc´ıaand ´ A.Rodr´ıguezPalacios Non-AssociativeNormedAlgebrasI:The Vidav–PalmerandGelfand–NaimarkTheorems
155C.F.DunklandY.Xu OrthogonalPolynomialsofSeveralVariables(SecondEdition)
156L.W.BeinekeandR.J.Wilson(eds.)withB.Toft TopicsinChromaticGraphTheory
157T.Mora SolvingPolynomialEquationSystemsIII:AlgebraicSolving
158T.Mora SolvingPolynomialEquationSystemsIV:BuchbergerTheoryandBeyond 159V.Berth ´ e andM.Rigo(eds.) Combinatorics,WordsandSymbolicDynamics
160B.Rubin IntroductiontoRadonTransforms:WithElementsofFractionalCalculusandHarmonic Analysis
161M.GherguandS.D.Taliaferro IsolatedSingularitiesinPartialDifferentialInequalities
162G.MolicaBisci,V.Radulescu,andR.Servadei VariationalMethodsforNonlocalFractional Problems
163S.Wagon TheBanach–TarskiParadox(SecondEdition)
164K.Broughan EquivalentsoftheRiemannHypothesisI:ArithmeticEquivalents
165K.Broughan EquivalentsoftheRiemannHypothesisII:AnalyticEquivalents
166M.BaakeandU.Grimm(eds.) AperiodicOrderII:CrystallographyandAlmostPeriodicity
167M.CabreraGarc´ıaand ´ A.Rodr´ıguezPalacios Non-AssociativeNormedAlgebrasII: RepresentationTheoryandtheZel’manovApproach
168A.Yu.Khrennikov,S.V.KozyrevandW.A.Z ´ u ˜ niga-Galindo UltrametricPseudodifferential EquationsandApplications
Non-AssociativeNormedAlgebras
Volume2:RepresentationTheoryand theZel’manovApproach
MIGUELCABRERAGARC ´ IA UniversidaddeGranada
´ ANGELRODR ´ IGUEZPALACIOS UniversidaddeGranada
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia
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©MiguelCabreraGarc´ıaand ´ AngelRodr´ıguezPalacios2018
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ISBN978-1-107-04306-0(hardback)
1.Banachalgebras.2.Algebra.I.Rodr´ıguezPalacios, ´ Angel.II.Title. QA326.C332014 512 .554–dc232013045718
ISBN978-1-107-04306-0Hardback
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ToAnaMar´ıaandIn ´ es
JBW ∗ -algebras, JB∗ -triplesrevisited,and aunit-freeVidav–Palmertypenon-associativetheorem
5.1.1Theresults
5.1.2Historicalnotesandcomments
5.2Preliminariesonanalyticmappings
5.2.1Polynomialsandhigherderivatives
5.2.4Historicalnotesandcomments
5.3Holomorphicautomorphismsofaboundeddomain
5.3.1Thetopologyofthelocaluniformconvergence53
5.3.2Holomorphicautomorphismsofaboundeddomain61
5.3.3TheCarath ´ eodorydistanceonaboundeddomain77
5.3.4Historicalnotesandcomments
5.4Completeholomorphicvectorfields
5.4.1LocallyLipschitzvectorfields
5.4.2Holomorphicvectorfields
5.4.3Completeholomorphicvectorfieldsand one-parametergroups
5.4.4Historicalnotesandcomments
5.5BanachLiestructuresforaut( ) andAut( )
5.5.1TherealBanachLiealgebraaut( )
5.5.2TherealBanachLiegroupAut( )
5.5.3Historicalnotesandcomments
5.6Kaup’sholomorphiccharacterizationof JB∗ -triples
5.6.1Boundedcirculardomains
5.6.2ThesymmetricpartofacomplexBanachspace187
5.6.3Numericalrangesrevisited
5.6.4ConcludingtheproofofKaup’stheorem
5.6.5Historicalnotesandcomments
5.7 JBW ∗ -triples
5.7.1Thebidualofa JB∗ -triple
5.7.2Themainresults
5.7.3Historicalnotesandcomments
5.8Operatorsintothepredualofa JBW ∗ -triple
5.8.1OnPełczy ´ nski’sproperty (V ∗ )
5.8.2 L-embeddedspaceshaveproperty (V ∗ )
5.8.3Applicationsto JB∗ -triples
5.8.4Historicalnotesandcomments
5.9Aholomorphiccharacterizationofnon-commutative JB∗ -algebras
5.9.1Completenormedalgebraswhosebidualsare non-commutative JB∗ -algebras
5.9.2Themainresult
5.9.3Historicalnotesandcomments
5.10Complementsonnon-commutative JB∗ -algebrasand JB∗ -triples
5.10.1Selectedtopicsinthetheoryofnon-commutative JBW ∗ -algebras
5.10.2Thestrong∗ topologyofa JBW ∗ -triple
5.10.3Isometriesofnon-commutative JB∗ -algebras
5.10.4Historicalnotesandcomments
6Representationtheoryfornon-commutative JB∗ -algebras andalternative C ∗ -algebras
6.1Themainresults
6.1.1Factorrepresentationsofnon-commutative JB∗ -algebras
6.1.2Associativityandcommutativityofnon-commutative JB∗ -algebras
6.1.3 JBW ∗ -factors
6.1.4Classifyingprime JB∗ -algebras:aZel’manovian approach
6.1.5Primenon-commutative JB∗ -algebrasarecentrally closed
6.1.6Non-commutative JBW ∗ -factorsandalternative W ∗ -factors 374
6.1.7Historicalnotesandcomments
6.2Applicationsoftherepresentationtheory
6.2.1Alternative C ∗ -and W ∗ -algebras
6.2.2Thestrongtopologyofanon-commutative JBW ∗ -algebra
6.2.3Primenon-commutative JB∗ -algebras
6.2.4Historicalnotesandcomments
6.3Afurtherapplication:commutativityofnon-commutative JB∗ -algebras
6.3.1LePage’stheorem,andsomenon-associative variants
6.3.2Themainresult
6.3.3Discussionofresultsandmethods
7Zel’manovapproach
7.1Classifyingprime JB∗ -triples
7.1.1Representationtheoryfor JB∗ -triples
7.1.2Buildingprime JB∗ -triplesfromprime C ∗ -algebras441
7.1.3Themainresults444
7.1.4Historicalnotesandcomments458
7.2AsurveyontheanalytictreatmentofZel’manov’sprime theorems461
7.2.1CompletenormedJ-primitiveJordanalgebras462
7.2.2Strong-versus-lightnormedversionsofthe Zel’manovprimetheorem467
7.2.3Thenormextensionproblem470
8Selectedtopicsinthetheoryofnon-associativenormedalgebras 477
8.1 H ∗ -algebras
8.1.1Preliminaries,andatheoremonpower-associative H ∗ -algebras
8.1.2Structuretheory
8.1.3Topologicallysimple H ∗ -algebrasare‘very’prime489
8.1.4Automaticcontinuity495
8.1.5Isomorphismsandderivationsof H ∗ -algebras502
8.1.6Jordanaxiomsforassociative H ∗ -algebras 508
8.1.7Realversuscomplex H ∗ -algebras 510
8.1.8Trace-classelementsin H ∗ -algebras 523
8.1.9Historicalnotesandcomments 545
8.2Extendingthetheoryof H ∗ -algebras:generalizedannihilator normedalgebras 561
8.2.1Themainresult 562
8.2.2Generalizedannihilatoralgebrasaremultiplicatively semiprime 571
8.2.3Generalizedcomplementednormedalgebras 577
8.2.4Historicalnotesandcomments 583
8.3Continuingthetheoryofnon-associativenormedalgebras586
8.3.1Continuityofhomomorphismsintonormedalgebras withouttopologicaldivisorsofzero 586
8.3.2CompletenormedJordanalgebraswithfinite J-spectrum 589
8.3.3Historicalnotesandcomments 595
8.3.4NormedJordanalgebrasafterAupetit’spaper[40]: asurvey
8.4Thejointspectralradiusofaboundedset
8.4.1Basicnotionsandresults
8.4.2Topologicallynilpotentnormedalgebras
8.4.3Involvingnearlyabsolute-valuedalgebras 638
8.4.4Involvingtensorproducts 642
8.4.5Historicalnotesandcomments 650
1.1.5Thecomplexificationofanormedrealalgebra
1.1.6Theunitalextensionandthecompletionofanormed
1.3.3Historicalnotesandcomments
1.4.3Discussingtheinclusion F(X , Y ) ⊆ K(X , Y ) inthe non-completesetting
1.4.4Historicalnotesandcomments
2.1.1Algebranumericalranges
2.1.2Operatornumericalranges
2.1.3Historicalnotesandcomments
2.2AnapplicationtoKadison’sisometrytheorem
2.2.1Non-associativeresults
2.2.2TheKadison–Paterson–Sinclairtheorem
2.2.3Historicalnotesandcomments
2.3TheassociativeVidav–Palmertheorem,startingfroma non-associativegerm
2.3.1Naturalinvolutionsof V -algebrasarealgebra involutions
2.3.2TheassociativeVidav–Palmertheorem
2.3.3Complementson C ∗ -algebras
2.3.4Introducingalternative C ∗ -algebras
2.3.5Historicalnotesandcomments
2.4 V -algebrasarenon-commutativeJordanalgebras
2.4.1Themainresult
2.4.2Applicationsto C ∗ -algebras
2.4.3Historicalnotesandcomments
2.5TheFrobenius–Zorntheorem,andthegeneralizedGelfand–Mazur–Kaplanskytheorem
2.5.1Introducingquaternionsandoctonions
2.5.2TheFrobenius–Zorntheorem 177
2.5.3ThegeneralizedGelfand–Mazur–Kaplanskytheorem192
2.5.4Historicalnotesandcomments
2.6Smooth-normedalgebras,andabsolute-valuedunitalalgebras203
2.6.1Determiningsmooth-normedalgebrasand absolute-valuedunitalalgebras 203
2.6.2Unit-freecharacterizationsofsmooth-normed algebras,andofabsolute-valuedunitalalgebras212
2.6.3Historicalnotesandcomments 216
2.7OtherGelfand–Mazurtypenon-associativetheorems 223
2.7.1Focusingoncomplexalgebras
2.7.2Involvingrealscalars
2.7.3Discussingtheresults
2.7.4Historicalnotesandcomments
2.8Complementsonabsolute-valuedalgebrasandalgebraicity249
2.8.1Continuityofalgebrahomomorphismsinto absolute-valuedalgebras 250
2.8.2Absolutevalueson H ∗ -algebras 251
2.8.3Freenon-associativealgebrasareabsolute-valued algebras 257
2.8.4Completenormedalgebraicalgebrasareofbounded degree 262
2.8.5Absolute-valuedalgebraicalgebrasare finite-dimensional 270
2.8.6Historicalnotesandcomments 274
2.9Complementsonnumericalranges 283
2.9.1Involvingtheuppersemicontinuityoftheduality mapping 284
2.9.2Theuppersemicontinuityofthepre-dualitymapping291
2.9.3Involvingthestrongsubdifferentiabilityofthenorm299
2.9.4Historicalnotesandcomments 310
3Concludingtheproofofthenon-associativeVidav–Palmertheorem 319
3.1Isometriesof JB-algebras 319
3.1.1Isometriesofunital JB-algebras 319
3.1.2Isometriesofnon-unital JB-algebras 324
3.1.3Ametriccharacterizationofderivationsof JB-algebras327
3.1.4 JB-algebraswhoseBanachspacesareconvex-transitive332
3.1.5Historicalnotesandcomments 336
3.2Theunitalnon-associativeGelfand–Naimarktheorem340
3.2.1Themainresult 340
3.2.2Historicalnotesandcomments 344
3.3Thenon-associativeVidav–Palmertheorem
3.3.1Themainresult
3.3.2Adualversion
3.3.3Historicalnotesandcomments 356
3.4Beginningthetheoryofnon-commutative JB∗ -algebras359
3.4.1 JB-algebrasversus JB∗ -algebras 359
3.4.2Isometriesofunitalnon-commutative JB∗ -algebras366
3.4.3Aninterlude:derivationsandautomorphismsof normedalgebras370
3.4.4Thestructuretheoremofisomorphismsof non-commutative JB∗ -algebras 381
3.4.5Historicalnotesandcomments 388
3.5TheGelfand–Naimarkaxiom a∗ a = a∗ a ,andthe non-unitalnon-associativeGelfand–Naimarktheorem392
3.5.1Quadraticnon-commutative JB∗ -algebras 393
3.5.2Theaxiom a∗ a = a∗ a onunitalalgebras397
3.5.3Aninterlude:thebidualandthespacialnumerical indexofanon-commutative JB∗ -algebra 404
3.5.4Theaxiom a∗ a = a∗ a onnon-unitalalgebras411
3.5.5Thenon-unitalnon-associativeGelfand–Naimark theorem414
3.5.6Vowden’stheorem
3.5.7Historicalnotesandcomments
3.6Jordanaxiomsfor C ∗ -algebras
3.6.1Jacobson’srepresentationtheory:preliminaries426
3.6.2Themainresult
3.6.3Jacobson’srepresentationtheorycontinued
3.6.4Historicalnotesandcomments
4Jordanspectraltheory
4.1InvolvingtheJordaninverse
4.1.1BasicspectraltheoryfornormedJordanalgebras451
4.1.2TopologicalJ-divisorsofzero
4.1.3Non-commutative JB∗ -algebrasare JB∗ -triples463
4.1.4ExtendingtheJordanspectraltheoryto Jordan-admissiblealgebras
4.1.5Theholomorphicfunctionalcalculusforcomplete normedunitalnon-commutativeJordancomplex algebras 480
4.1.6Acharacterizationofsmooth-normedalgebras487
4.1.7Historicalnotesandcomments 490
4.2Unitariesin JB∗ -triplesandinnon-commutative JB∗ -algebras497
4.2.1AcommutativeGelfand–Naimarktheoremfor JB∗ -triples 498
4.2.2Themainresults 505
4.2.3Russo–Dyetypetheoremsfornon-commutative JB∗ -algebras 518
4.2.4Atouchofreal JB∗ -triplesandofrealnoncommutative JB∗ -algebras 521
4.2.5Historicalnotesandcomments 527
4.3 C ∗ -and JB∗ -algebrasgeneratedbyanon-self-adjoint idempotent 536
4.3.1Thecaseof C ∗ -algebras 536
4.3.2Thecaseof JB∗ -algebras 552
4.3.3Anapplicationtonon-commutative JB∗ -algebras560
4.3.4Historicalnotesandcomments562
4.4Algebranormsonnon-commutative JB∗ -algebras 565
4.4.1TheJohnson–Aupetit–Ransforduniqueness-of-norm theorem 566
4.4.2Anon-completevariant 571
4.4.3Themainresults 573
4.4.4Theuniqueness-of-normtheoremforgeneral non-associativealgebras 577
4.4.5Historicalnotesandcomments 592
4.5 JB∗ -representationsandalternative C ∗ -representationsof hermitianalgebras604
4.5.1Preliminaryresults605
4.5.2Themainresults611
4.5.3Aconjectureonnon-commutative JB∗ -equivalent algebras630
4.5.4Historicalnotesandcomments632
4.6Domainsofclosedderivations636
4.6.1Stabilityundertheholomorphicfunctionalcalculus636
4.6.2Stabilityunderthegeometricfunctionalcalculus644
4.6.3Historicalnotesandcomments665
References–Papers
Preface
Thecoreofthebookrevisited
IntheprefacetoVolume1weproposedasthe‘leitmotiv’ofourworktoremoveassociativityintheabstractcharacterizationsofunital(associative) C ∗ -algebrasgiven eitherbytheGelfand–NaimarktheoremorbytheVidav–Palmertheorem,andto study(possiblynon-unital)closed ∗-subalgebrasoftheGelfand–NaimarkorVidav–Palmeralgebrasbornafterremovingassociativity.
Tobemoreprecise,foranorm-unitalcompletenormed(possiblynon-associative) complexalgebra A,weconsideredthefollowingconditions:
(GN)(Gelfand–Naimarkaxiom). Thereisaconjugate-linearvectorspaceinvolution ∗ onAsatisfying 1∗ = 1 and a∗ a = a 2 foreveryainA.
(VP)(Vidav–Palmeraxiom). A = H (A, 1) + iH (A, 1).
Inbothconditions, 1 denotestheunitof A,whereas,in(VP), H (A, 1) standsforthe closedrealsubspaceof A consistingofthoseelements h ∈ A suchthat f (h) belongs to R foreveryboundedlinearfunctional f on A satisfying f = f (1) = 1.
Contrarytowhathappensintheassociativecase[696,725,787,930],inthenonassociativesetting,(GN)and(VP)arenotequivalentconditions.Indeed,asproved inLemma2.2.5,itiseasilyseenthat(GN)implies(VP),but,asshownbyExample 2.3.65,theconverseimplicationisnottrue.Therefore,afterintroducing‘alternative C ∗ -algebras’and‘non-commutative JB∗ -algebras’,andrealizingthattheformerare particularcasesofthelatter,wespecifiedhow,bymeansofTheoremsGNandVP whichfollow,thebehaviouroftheGelfand–NaimarkandtheVidav–Palmeraxioms inthenon-associativesettingareclarified.
TheoremGN Norm-unitalcompletenormedcomplexalgebrasfulfillingtheGelfand–NaimarkaxiomarenothingotherthanunitalalternativeC ∗ -algebras.
TheoremVP Norm-unitalcompletenormedcomplexalgebrasfulfillingtheVidav–Palmeraxiomarenothingotherthanunitalnon-commutativeJB∗ -algebras.
Preface
ThenweannouncedasthemaingoalofourworktoproveTheoremsGNandVP, togetherwiththeirunit-freevariants,andto‘describe’alternative C ∗ -algebrasand non-commutative JB∗ -algebrasbymeansoftheso-calledrepresentationtheory. SinceTheoremsGNandVPandtheunit-freevariantofTheoremGNwerealready provedinTheorems3.2.5,3.3.11,and3.5.53,respectively,itremainsthemain objectiveofourworktoprovetheunit-freevariantofTheoremVP,andtodevelopthe representationtheoryofalternative C ∗ -algebrasandnon-commutative JB∗ -algebras. WenowdothisinChapters5and6respectively.Indeed,theunit-freevariantof TheoremVPisprovedinTheorem5.9.9,whereastherepresentationtheoryof alternative C ∗ -algebrasandnon-commutative JB∗ -algebrascanbesummarizedby meansofCorollaries6.1.11and6.1.12,Theorem6.1.112,andCorollary6.1.115.
ThecontentofVolume2
AswecommentedintheprefaceofVolume1,thedividinglinebetweenthetwo volumescouldbedrawnbetweenwhatcanbedonebeforeandafterinvolvingthe holomorphictheoryof JB∗ -triplesandthestructuretheoryofnon-commutative JB∗algebras.ThenthecontentofVolume1wasdescribedinsomedetail,andatentative contentofVolume2wasoutlined.Nowwearegoingtospecifywithmoreprecision thecontentofthepresentsecondvolume.
Chapter5
ThemaingoalofthisfirstchapterofVolume2istoprovewhatcanbeseenas aunit-freeversionofthenon-associativeVidav–Palmertheorem,namelythat noncommutativeJB∗ -algebrasarepreciselythosecompletenormedcomplexalgebras havinganapproximateunitboundedbyone,andwhoseopenunitballisahomogeneousdomain [365](seeTheorem5.9.9).Someingredientsinthelongproofofthis resultwerealreadyestablishedinVolume1.ThisisthecaseoftheBohnenblust–KarlinCorollary2.1.13,thenon-associativeVidav–PalmertheoremprovedinTheorem3.3.11aswellasitsdualversionshowninCorollary3.3.26,Proposition3.5.23 (thateverynon-commutative JB∗ -algebrahasanapproximateunitboundedbyone), Theorem4.1.45(thatnon-commutative JB∗ -algebrasare JB∗ -triplesinanatural way),andtheequivalence(ii)⇔(vii)intheBraun–Kaup–UpmeierTheorem4.2.24. ♣ Thenewrelevantingredientswhichareprovedinthechapterarethefollowing:
(i)Edward’sfundamentalFact5.1.42,whichdescribeshow JBW -algebrasand JBW ∗ -algebrasaremutuallydetermined,andimplies,via[738],theuniquenessofthepredualofanynon-commutative JBW ∗ -algebra(seeTheorem 5.1.29(iv)).
(ii)TheKaup–Stach ´ ocontractiveprojectiontheoremfor JB∗ -triples(seeTheorem 5.6.59).
(iii)Kaup’sholomorphiccharacterizationof JB∗ -triplesasthosecomplexBanach spaceswhoseopenunitballisahomogeneousdomain(seeTheorem5.6.68).
(iv)Dineen’scelebratedresultthatthebidualofa JB∗ -tripleisa JB∗ -triple(see Proposition5.7.10).
(v)TheBarton–Horn–Timoneybasictheoryof JBW ∗ -triplesestablishingtheseparate w∗ -continuityofthetripleproductofagiven JBW ∗ -triple(seeTheorem 5.7.20)andtheuniquenessofthepredual(seeTheorem5.7.38).
(vi)TheBarton–Timoneytheoremthatthepredualofany JBW ∗ -tripleis L-embedded(seeTheorem5.7.36).
(vii)TheChu–Iochum–Loupiasresultthatboundedlinearoperatorsfroma JB∗tripletoitsdualareweaklycompact(seeCorollary5.8.33)or,equivalently, thatallcontinuousproductsontheBanachspaceofa JB∗ -tripleareArens regular(seeFact5.8.39).
Theoriginalreferencesfortheresultsjustlistedare[222],[382,597],[381],[213], [854,979],[854],and[172],respectively.Ourproofoftheseresultsarenotalways theoriginalones,althoughsometimesthelatterunderlietheformer.Thisisthecase ofresults(ii)and(iii),whichinourdevelopmentdependonthefoundationsofthe infinite-dimensionalholomorphydonein[710,751,814,837,1113,1114,1124] (seeSections5.2to5.6),onthedesignofproofsuggestedin[710,Section2.5],and, attheend,onnumericalrangetechniquesincludedinSubsection5.6.3.Ontheother hand,ourproofofresult(v)isnew,and,contrarytowhathappensintheoriginal one,itavoidsanyBanachspaceresultonuniquenessofpreduals.Indeed,ourproof ofTheorem5.7.20involvesonlyresult(ii)andtheBarton–TimoneyTheorem5.7.18, whereasourproofofTheorem5.7.38dependsonlyonTheorem5.7.20(whoseproof hasbeenjustremarkedon),result(i),andHorn’sCorollary5.7.28(i)(b).
Concerningresult(vii),itisnoteworthythatamuchfinertheoremisprovedin [172].Namely,thateveryboundedlinearoperatorfroma JB∗ -tripletoitsdualfactors throughacomplexHilbertspace.Theproofofthismoregeneraltheorem(asketch ofwhichcanbefoundin§5.10.151)isveryinvolved,andshallnotbecompletely discussedinourwork.Asamatteroffact,were-encounterresult(vii)bycombining results(iv)and(vi)withCorollary5.8.19(assertingthat,if Y isaBanachspacesuch that Y hasproperty(V ∗ ),theneveryboundedlinearoperatorfrom Y to Y isweakly compact)andTheorem5.8.27(that L-embeddedBanachspaceshaveproperty(V ∗ )). Corollary5.8.19andTheorem5.8.27justreviewedareduetoGodefroy–Iochum [957]andPfitzner[1044],respectively.Nevertheless,theproofofCorollary5.8.19 in[957]reliesheavilyonProposition5.8.14,whoseargumentshavebeenlostin theliterature(see§5.8.42).OurproofofProposition5.8.14istakenfromPfitzner’s privatecommunication[1047].
OncethemainobjectiveofthechapterisreachedinSection5.9,thechapter concludeswithasectiondevotedtosomecomplementsonnon-commutative JB∗algebrasand JB∗ -triples.
InSubsection5.10.1weintroducethestrong∗ topologyofanon-commutative JBW ∗ -algebra[19]andapplyittobuildupafunctionalcalculusateachnormal element a ofanon-commutative JBW ∗ -algebra A,whichextendsthecontinuous
functionalcalculus(cf.Corollary4.1.72)andhasasenseforallreal-valuedbounded lowersemicontinuousfunctionsonJ-sp(A, a).Thenwefollow[366]toprovea variantfornon-commutative JBW ∗ -algebrasofKadison’sisometrytheoremfor unital C ∗ -algebras(cf.Theorem2.2.29),aconsequenceofwhichisthatlinearly isometricnon-commutative JBW ∗ -algebrasareJordan-∗-isomorphic.(Werecall thatlinearlyisometric(possiblynon-unital) C ∗ -algebrasareJordan-∗-isomorphic(a consequenceofTheorem2.2.19),butthatlinearlyisometric(evenunital)noncommutative JB∗ -algebrasneednotbeJordan-∗-isomorphic(cf.Antitheorem 3.4.34).)Wealsoprovethegeneralizationtonon-commutative JBW ∗ -algebras ofAkemann’stheorem[826]assertingthecoincidenceofthestrong∗ andMackey topologiesonboundedsubsetsofany W ∗ -algebra.
InSubsection5.10.2weintroduceandstudythestrong∗ topologyofa JBW ∗tripleasdonebyBartonandFriedman[853,60],andfollow[1061]toprovethat, whenanon-commutative JBW ∗ -algebraisviewedasa JBW ∗ -triple,itsnew(triple) strong∗ topologycoincideswiththe(algebra)strong∗ topologyintroducedinSubsection5.10.1.WealsoproveZizler’srefinement[1137]ofLindenstrauss’stheorem [1001]onnorm-densityofoperatorswhosetransposeattaintheirnorm,andapply ittoproveavariantfor JBW ∗ -triplesoftheso-calledlittleGrothendieck’stheorem [853,964,1040,1052].
InSubsection5.10.3weprovidethereaderwithafullnon-associativediscussion oftheKadison–Paterson–SinclairTheorem2.2.19onsurjectivelinearisometries of(possiblynon-unital) C ∗ -algebras[366].Tothisendweintroducethemultiplier non-commutative JB∗ -algebra M (A) ofagivennon-commutative JB∗ -algebra A, andprovethat M (A) coincideswiththe JB∗ -tripleofmultipliers[873]ofthe JB∗tripleunderlying A.ThenwealsoprovethattheKadison–Paterson–Sinclairtheorem remainstrueverbatimforsurjectivelinearisometriesfromnon-commutative JB∗algebrastoalternative C ∗ -algebras,andthatnofurtherverbatimgeneralizationis possible.
Chapter6
Implicitly,therepresentationtheoryof JB-algebrasunderliesourworksince,without providingthereaderwithaproof,wetookfromtheHanche-Olsen–Stormerbook [738]theverydeepfactthattheclosedsubalgebraofa JB-algebrageneratedbytwo elementsisa JC -algebra(cf.Proposition3.1.3).Inthatwaywewereabletodevelop thebasictheoryofnon-commutative JB∗ -algebras(includingthenon-associative Vidav–PalmerTheorem3.3.11andWright’sfundamentalFact3.4.9whichdescribes how JB-algebrasand JB∗ -algebrasaremutuallydetermined)withoutanyfurther implicitorexplicitreferencetorepresentationtheory.Infact,weavoidedanydependenceonrepresentationtheorythroughoutallofVolume1,andtotheendofChapter 5ofthepresentvolume.
Now,inChapter6,weconcludethebasictheoryofnon-commutative JB∗algebras,andfollow[19,124,125,222,481,482,641]todevelopindepththe
Preface xxi
representationtheoryofnon-commutative JB∗ -algebrasand,inparticular,that ofalternative C ∗ -algebras.Tothisend,inSubsection6.1.1weintroducenoncommutative JBW ∗ -factorsandnon-commutative JBW ∗ -factorrepresentationsofa givennon-commutative JB∗ -algebra,andprovethateverynon-commutative JB∗algebrahasafaithfulfamilyoftypeInon-commutative JBW ∗ -factorrepresentations. Whentheseresultsspecializeforclassical C ∗ -algebras,typeInon-commutative JBW ∗ -factorsarenothingotherthanthe(associative) W ∗ -factorsconsistingofall boundedlinearoperatorsonsomecomplexHilbertspace[738,Proposition7.5.2], and,consequently,typeI W ∗ -factorrepresentationsofa C ∗ -algebra A areprecisely theirreduciblerepresentationsof A oncomplexHilbertspaces.Subsection6.1.2 dealswithafirstapplicationoftherepresentationtheoryoutlinedabove,which allowsustoshowthatnon-commutative JB∗ -algebrasareassociativeandcommutativeif(andonlyif)theyhavenononzeronilpotentelement.Asaconsequence, weobtainthatalternative C ∗ -algebrasarecommutativeifandonlyiftheyhave nononzeronilpotentelement[340].ThisgeneralizesKaplansky’sassociative forerunner[761,TheoremBinAppendixIII].InSubsection6.1.3,weinvolvethe theoryof JB-algebras[738],andinvokeresult(i)in ♣ toclassifyall(commutative) JBW ∗ -factors.Thisclassificationisappliedtoprovethati-special JB∗ -algebras are JC ∗ -algebras.InSubsection6.1.4,wecombinetheresultjustreviewedwith Zel’manoviantechniques[437,662]toprovethat,if J isaprime JB∗ -algebra,and if J isneitherquadratic(cf.Corollary3.5.7)norequaltotheunique JB∗ -algebra whoseself-adjointpartis H3 (O) (cf.Example3.1.56andTheorem3.4.8),then eitherthereexistsaprime C ∗ -algebra A suchthat J isaclosed ∗-subalgebraof the JB∗ -algebra M (A)sym containing A,orthereexistsaprime C ∗ -algebra A with a ∗-involution τ suchthat J isaclosed ∗-subalgebraof M (A)sym containedin H (M (A), τ) andcontaining H (A, τ) [255].InSubsection6.1.5,weintroducetotally primenormedalgebrasandultraprimenormedalgebras,andprovethattotallyprime normedcomplexalgebrasarecentrallyclosed,andthatultraprimenormedalgebras aretotallyprime[149].Thenwecombinetheclassificationtheoremofprime JB∗ -algebrasreviewedabovewiththefactthatprime C ∗ -algebrasareultraprime [1012]toshowthatprimenon-commutative JB∗ -algebrasareultraprime,andhence centrallyclosed.InSubsection6.1.6,wecombinethecentralclosednessofprime non-commutative JB∗ -algebraswithatopologicalreadingofMcCrimmon’spaper [436]toprovethatnon-commutative JBW ∗ -factorsareeithercommutativeorsimple quadraticoroftheform B(λ) forsome(associative) W ∗ -factor B andsome0 ≤ λ ≤ 1. ThistheoremisoriginallyduetoBraun[124].Asaconsequence,alternative W ∗factorsareeitherassociativeorequaltothealternative C ∗ -algebraofcomplex octonions(cf.Proposition2.6.8).
NowthatwehavereviewedSection6.1indetail,wewillexplainthecontentof theremainingsectionsofChapter6.Section6.2dealswiththemainapplications oftherepresentationtheory,namelythestructureofalternative C ∗ -algebras[125, 331,481],thedefinitionandpropertiesofthestrongtopologyofanon-commutative JBW ∗ -algebra[482],andtheclassificationofprimenon-commutative JB∗ -algebras
[363].Finally,Section6.3dealswitharatherincidentalapplication.Indeed,we follow[860]toproveaLePagetypetheoremfornon-commutative JB∗ -algebras, anddiscussLePage’stheorem[999]inageneralnon-associativeandnon-star setting.
Chapter7
ThischapterdealswiththeanalytictreatmentofZel’manov’sprimetheoremsfor Jordanstructures,thuscontinuingtheapproachbeguninSubsection6.1.4.
InSubsection7.1wefollow[448,449]toproveasthemainresultthat,if X isa prime JB∗ -triplewhichisneitheranexceptionalCartanfactornoraspintriplefactor, theneitherthereexistaprime C ∗ -algebra A andaself-adjointidempotent e inthe C ∗algebra M (A) ofmultipliersof A suchthat X isaclosedsubtripleof M (A) contained in eM (A)(1 e) andcontaining eA(1 e),orthereexistaprime C ∗ -algebra A,aselfadjointidempotent e ∈ M (A),anda ∗-involution τ on A with e + eτ = 1 suchthat X isaclosedsubtripleof M (A) containedin H (eM (A)eτ , τ) andcontaining H (eAeτ , τ)
Amongthemanytoolsinvolvedintheproofoftheaboveclassificationtheorem, weemphasizeHorn’sdescriptionofCartanfactors[330],thecoreoftheproofof Zel’manov’sprimetheoremforJordantriples[663,1133,1134],andthecomplementaryworkbyD’AmourandMcCrimmononthetopic[920,921].Proofsofthese toolsarenotdiscussedinourdevelopment.ThemainresultsintheFriedman–Russo paper[270],whoseproofsareoutlinedinourdevelopment,arealsoinvolved.Itis noteworthythat,throughthedescriptionofprime JB∗ -algebrasprovedinSubsection 6.1.4,Zel’manov’sworkunderliesagaintheproofoftheclassificationtheoremof prime JB∗ -tripleswearedealingwith.
InSection7.2wesurveyindetailotherapplicationsofZel’manov’sprimetheoremsonJordanstructurestothestudyofnormedJordanalgebrasandtriples.
InSubsection7.2.1weincludethegeneralcompletenormedversion[146]ofthe Anquela–Montaner–Cort ´ es–SkosyrskiiclassificationtheoremofJ-primitiveJordan algebras[21,585],aswellasthemorepreciseclassificationtheoremofJ-primitive JB∗ -algebras[255,525].
InSubsection7.2.2weincludestructuretheoremsforsimplenormedJordan algebras[151](seealso[539])andnon-degeneratelyultraprimecompletenormed Jordancomplexalgebras[152](seealso[428,855]).Thissubsectiondealsalsowith thelimitsofnormedversionsofZel’manovprimetheorems,aquestionwhichwas firstconsideredin[893],andculminatesinthepaperofMoreno,Zel’manov,and theauthors[147]whereitisprovedthatanassociativepolynomial p over K isa Jordanpolynomialifandonlyif,foreveryalgebranorm · ontheJordanalgebra M∞ (K)sym ,theactionof p on M∞ (K) is · -continuous(seealso[447,1082]).
Subsection7.2.3dealswiththeso-callednormextensionproblem,whichinits rootsiscruciallyrelatedtonormedversionsofZel’manov’sprimetheorems.Thefirst significativeprogressonthisproblem(reviewedofcourseinthissubsection)is duetoRodr´ıguez,Slinko,andZel’manov[538],whoasthemainresultprovethat,
Preface
if A isarealorcomplexassociativealgebrawithlinearalgebrainvolution ∗,if A isa‘∗-tightenvelopeof H (A, ∗)’,iftheJordanalgebra H (A, ∗) issemiprime,andif · isacompletealgebranormon H (A, ∗),thenthereexistsanalgebranormon A whoserestrictionto H (A, ∗) isequivalentto · .TheappropriateversionsforJordan triplesoftheresultsof[538],duetoMoreno[1025,1026],arealsoincluded.The subsectionconcludeswithafulldiscussionofresultsonthenormextensionproblem inageneralnon-associativesetting.Themainreferenceforthistopicis[1029].Other relatedresultsin[1027,1059,1064]arealsoreviewed.
Chapter8
Wedevotethisconcludingchaptertodevelopingsomeofourfavouriteparcelsofthe theoryofnon-associativenormedalgebras,notpreviouslyincludedinourwork.
Thefirstsectionofthechapterdealswith H ∗ -algebras,incidentallyintroducedin Volume1ofourwork.Thereasonablywell-behavedco-existenceoftwostructures, namelythatofanalgebraandthatofaHilbertspace,becomestheessenceofsemiH ∗ -algebras.Indeed,theyarecompletenormedalgebras A endowedwitha(vector space)conjugate-linearinvolution ∗,andwhosenormderivesfromaninnerproduct insuchawaythat,foreach a ∈ A,theadjointoftheleftmultiplication La isprecisely La∗ ,andtheadjointoftherightmultiplication Ra is Ra∗ .SinceAmbrose’spioneering paper[20],itiswell-knownthatassociativesemi-H ∗ -algebraswithzeroannihilator are H ∗ -algebras,i.e.theirinvolutionsarealgebrainvolutions.Butthisisnolonger trueingeneral.
WebeginSubsection8.1.1byrecallingthoseresultsonsemi-H ∗ -algebras,which werealreadyprovedinVolume1ofourwork.Thenweintroducetheclassical topologicallysimpleassociativecomplex H ∗ -algebra HS (H ) ofallHilbert–SchmidtoperatorsonanonzerocomplexHilbertspace H ,andshowhowthis algebraallowsustoconstructnaturalexamplesofJordanandLie H ∗ -algebras. Aftershowinghowthenormofasemi-H ∗ -algebrawithzeroannihilatordetermines itsinvolution,weprovethatpower-associative H ∗ -algebrasarenon-commutative Jordanalgebras[714].
InSubsection8.1.2,weestablishtwofundamentalstructuretheoremsforasemiH ∗ -algebra A,which,intwosuccessivesteps,reducethegeneralcasetotheonethat A haszeroannihilator,andthecasethat A haszeroannihilatortotheonethat A is topologicallysimple[199].
Accordingtothestructuretheorycommentedintheprecedingparagraph,topologicallysimplesemi-H ∗ -algebrasmeritbeingstudiedindepth.Thisisdonein Subsection8.1.3.Tothisendweintroducetotallymultiplicativelyprimenormed algebras,showthattheyaretotallyprime,andprovethattopologicallysimplecomplexsemi-H ∗ -algebrasaretotallymultiplicativelyprime[889].Since,aswealready commentedinourreviewofSubsection6.1.5,totallyprimenormedcomplexalgebrasarecentrallyclosed,itfollowsthattopologicallysimplecomplex H ∗ -algebras arecentrallyclosed[148,149].
Preface
Thecentralclosednessoftopologicallysimplecomplex H ∗ -algebrasjustreviewed becomesthekeytoolofSubsection8.1.4,whereweprovethatderivationsofcomplexsemi-H ∗ -algebraswithzeroannihilatorarecontinuous[624],andthatdenserangealgebrahomomorphismsfromcompletenormedcomplexalgebrastocomplex H ∗ -algebraswithzeroannihilatorarealsocontinuous[526].
InSubsection8.1.5weshowthatisomorphiccomplex H ∗ -algebraswithzeroannihilatorare ∗-isomorphic,andthatbijectivealgebra ∗-homomorphismsbetweentopologicallysimplecomplex H ∗ -algebrasarepositivemultiplesofisometries(hence, essentially,atopologicallysimplecomplex H ∗ -algebrahasaunique H ∗ -algebra structure)[198].Theseresultsfollowfromastructuretheoremforbijectivealgebrahomomorphismsbetweencomplex H ∗ -algebraswithzeroannihilator,which becomestheappropriate H ∗ -variantofthestructuretheoremforbijectivealgebra homomorphismsbetweennon-commutative JB∗ -algebrasprovedinTheorem3.4.75.
InSubsection8.1.6,weprovetheappropriate H ∗ -variantoftheJordancharacterizationof C ∗ -algebrasestablishedinTheorem3.6.30[518].Amorethansatisfactory H ∗ -variantofTheorem3.6.25isalsoobtained[624].
Subsection8.1.7isdevotedtoprovidinguswiththeappropriatetoolstotransfer resultsfromcomplexsemi-H ∗ -algebrastorealones.Thebasictoolassertsthat thecomplexificationofanyreal(semi-)H ∗ -algebrabecomesacomplex(semi-)H ∗algebrainanaturalway.Thisquiteelementaryfactalreadyallowstoconvertmany complexresultsintorealones,alloftheminvolvingtheassumptionthatthealgebra haszeroannihilator.Thetreatmentoftopologicallysimplereal(semi-)H ∗ -algebras ismoreelaborated:therearenotopologicallysimplereal(semi-)H ∗ -algebrasother thantopologicallysimplecomplex(semi-)H ∗ -algebras,regardedasrealalgebras, andthereal(semi-)H ∗ -algebrasofallfixedpointsforaninvolutiveconjugate-linear algebra ∗-homomorphismonatopologicallysimplecomplex(semi-)H ∗ -algebra [142].Thisreductionoftopologicallysimplereal(semi-) H ∗ -algebrastocomplex onesallowsustotransfertheremainingresultsknowninthecomplexsettingtothe realsetting.Inparticular,weprovethatdense-rangealgebrahomomorphismsfrom H ∗ -algebraswithzeroannihilatortotopologicallysimple H ∗ -algebrasaresurjective. Then,afterintroducing H ∗ -idealsofanarbitrarynormed ∗-algebra,weprovethat topologicallysimplenormed ∗-algebrashaveatmostone H ∗ -ideal[687].
WebeginSubsection8.1.8byintroducingthecompletenormedcomplex ∗-algebra (TC (H ), · τ ) ofalltrace-classoperatorsonacomplexHilbertspace H ,aswell asthe · τ -continuoustrace-formonit.Thenweshowthat (TC (H ), · τ ) canbe intrinsicallydeterminedintothe H ∗ -algebra (HS (H ), · ) ofallHilbert–Schmidt operatorson H .Thisfactallowsustoreplace HS (H ) withanarbitraryrealor complex(possiblynon-associative)semi-H ∗ -algebra A withzeroannihilator,tobuild anappropriatesubstituteof (TC (H ), · τ ) into A,denotedby (τ c(A), · τ ),and todiscusswhetherornota · τ -continuoustrace-formon τ c(A) doesexist.We provethat,forasemi-H ∗ -algebra A withzeroannihilator, τ c(A) isa ∗-invariant idealof A, (τ c(A), · τ ) isbothanormedalgebraandadualBanachspace,andthe existenceofa · τ -continuoustrace-formon τ c(A) dependsontheexistenceof
Preface
an‘operator-bounded’approximateunitin A [424].This,togetherwithdeepresults establishedinVolume1(namelyTheorem3.5.53andProposition4.5.36(ii)),allows ustoprovethatacomplex H ∗ -algebra A withzeroannihilatorisalternativeifand onlyif (A, · ) hasanapproximateunitoperator-boundedby1,andthepredualof (τ c(A), · τ ) isanon-associative C ∗ -algebra.
IntheconcludingSubsection8.1.9,wesurveytheclassificationtheoremsof topologicallysimple H ∗ -algebrasinthemostfamiliarclassesofalgebras.Thus, startingfromthewell-knownfactthattherearenotopologicallysimpleassociativecomplex H ∗ -algebrasotherthanthoseoftheform HS (H ) foranonzero complexHilbertspace H [20,374],thecorrespondingtheoremsfortopologically simplealternative[1042],Jordanandnon-commutativeJordan[199,1118,1119], Lie[197,460,687],Malcev[141],orstructurable[140,144] H ∗ -algebrasare established.
Section8.2dealswithgeneralizedannihilatornormedalgebras,whichbecome non-stargeneralizationsof H ∗ -algebraswithzeroannihilator.Weprovethatany generalizedannihilatorcompletenormedrealorcomplexalgebrawithzeroweak radical(cf.Definition4.4.39)istheclosureofthedirectsumofitsminimal closedideals,whichareindeedtopologicallysimplenormedalgebras[259]. Wealsoshowthattheweakradicalofanyrealorcomplexsemi-H ∗ -algebra coincideswithitsannihilator,sothatthestructuretheoremforsemi-H ∗ -algebras withzeroannihilatorprovedinSubsection8.1.2isrediscovered.Weintroduce multiplicativelysemiprimealgebras(i.e.algebrassuchthatboththeyandtheir multiplicationalgebrasaresemiprime),andshowthatgeneralizedannihilator normedalgebrasaremultiplicativelysemiprime[876].Evenmore,wecharacterize generalizedannihilatornormedalgebrasamongthosenormedalgebraswhich aremultiplicativelysemiprime.Weintroducegeneralizedcomplementednormed algebras,whichareparticularcasesofgeneralizedannihilatornormedalgebras,and provethat,if A isageneralizedcomplementedcompletenormedalgebrawithzero weakradical,andif {Ai }i∈I standsforthefamilyofitsminimalclosedideals,then foreach a ∈ A thereexistsauniquesummablefamily {ai }i∈I in A suchthat ai ∈ Ai forevery i ∈ I ,and a = i∈I ai [259,846].
Section8.3dealswithothercomplementsintothetheoryofnon-associative normedalgebras.InSubsection8.3.1weprovethatalgebrahomomorphismsfrom completenormedcomplexalgebrastocompletenormedcomplexalgebraswithno nonzerotwo-sidedtopologicaldivisorofzeroarecontinuous[529].InSubsection 8.3.2weshowthatcompletenormedJ-semisimplenon-commutativeJordancomplex algebras,eachelementofwhichhasafiniteJ-spectrum,areafinitedirectsumof closedsimpleidealswhichareeitherfinite-dimensionalorquadratic,andderive thatcompletenormedsemisimplealternativecomplexalgebras,eachelementof whichhasafinitespectrum,arefinite-dimensional[91].Aftertheusualsubsection devotedtohistoricalnotesandcomments,weincludeacomprehensivesurveyonthe moresignificantresultsonnormedJordanalgebraswhichhavebeennotpreviously developedinourwork.
TheconcludingSection8.4dealswiththenon-associativediscussiondonein [452,453]oftheRota–Strangpaper[544](soinparticularofProposition4.5.2,cf. p.632ofVolume1),andofthetheoryoftopologicallynilpotentnormed(associative)algebrasdevelopedin[927,928,929,1020](seealso[786,pp.515–7],[1156, Section11],and§8.4.121).Thesectiondiscussesalsonon-associativeversionsof relatedresultspublishedin[569,615,1083](seealso[1030]),andincorporates proofsofmostauxiliaryresultsinvokedbutnotprovedin[452,453].Amongthese proofs,weemphasizethatofTheorem8.4.76,courtesyofShulmanandTurovskii.
InSubsection8.4.1weintroducethenotionof(joint)spectralradius r(S) ofa boundedsubset S ofanynormedalgebra A.Thenweproveoneofthekeyresults inthewholesection,namelythat, ifAisanormedalgebra,andifSisabounded subsetofAwith r(S)< 1,thenthemultiplicativelyclosedsubsetofAgeneratedbyS isbounded,andhasthesamespectralradiusasS.
InSubsection8.4.2,weintroducetopologicallynilpotentnormedalgebrasasthose normedalgebraswhoseclosedunitballshavezerospectralradius.Amongtheresults obtained,weemphasizethefollowing:
(i)Anormedassociativealgebra A istopologicallynilpotentifandonlyifsoisthe normedJordanalgebra Asym obtainedbysymmetrizationofitsproduct.
(ii)Everynon-topologicallynilpotentnormedalgebracanbeequivalentlyalgebrarenormedinsuchawaythatthespectralradiusofthecorrespondingclosedunit ballisarbitrarilycloseto1.
(iii)Everytopologicallynilpotentcompletenormedalgebraisequaltoitsweak radical.
InSubsection8.4.3,weshowthat,foreverymember A inalargeclassofnormed algebras(whichcontainsallcommutative C ∗ -algebras,all JB-algebras,andall absolute-valuedalgebras),theconclusioninProposition4.5.2hasthefollowing strongerform:foreachboundedandmultiplicativelyclosedsubset S of A wehave thatsup{ s : s ∈ S}≤ 1.
InSubsection8.4.4,weinvolveinourdevelopmenttensorproductsofalgebras. Thusweprovethattheprojectivetensorproductoftwonormedalgebrasistopologicallynilpotentwheneversomeofthemaretopologicallynilpotent,andthatinfact theconverseistruewheneversomeofthemareassociative.Moreover,associativity intheaboveconversecannotberemoved.Wealsoprovethatanormedalgebra A is topologicallynilpotentifandonlyifsoisthenormedalgebra C0 (E , A) forsome (equivalently,every)Hausdorfflocallycompacttopologicalspace E .Theresults obtainedabouttensorproductsofnormedalgebrasarethenappliedtoshowthat mostnotionsintroducedinthesectioncanbenon-triviallyexemplifiedintoaclass ofalgebrasalmostarbitrarilyprefixed.
Onthehistoricalnotes
AsinVolume1,eachsectionofthepresentvolumeconcludeswithasubsection devotedtohistoricalnotesandcomments.ParaphrasingDinnen[1155,p.X],inthese
Preface
notes‘weprovideinformationonthehistoryofthesubjectandreferencesforthe materialpresented.Wehavetriedtobeascarefulaspossibleinthisregardand takeresponsibilityfortheinevitableerrors.Accurateandcomprehensiverecordsof thiskindarenotaluxurybutessentialbackgroundinformationinappreciatingand understandingasubjectanditsevolution’.
Errata: AlistoferrataforVolume1canbefoundinthewebpageofVolume2: www.cambridge.org/9781107043114.WehopetocontinuethisforbothVolumes. Pleasesendcorrectionsto:cabrera@ugr.esand/orapalacio@ugr.es.
Acknowledgements
Weareindebtedtomanymathematicianswhoseencouragement,questions,suggestions,andkindly-sentreprintsgreatlyinfluenceduswhilewewerewritingthis work:M.D.Acosta,J.Alaminos,C.Aparicio,R.M.Aron,B.A.Barnes,J.Becerra, H.Behncke,G.Benkart,A.Browder,M.J.Burgos,J.C.Cabello,A.J.Cabrera, A.Ca ˜ nada,C.-H.Chu,M.J.Crabb,J.Cuntz,C.M.Edwards,F.J.Fern ´ andezPolo, J.E.Gal ´ e,E.Garc´ıa,G.Godefroy,M.G ´ omez,A.Y.Helemskii,R.Iordanescu, V.Kadets,O.Loos,G.L ´ opez,J.Mart´ınezMoreno,M.Mathieu,C.M.McGregor, P.Mellon,A.Morales,J.C.NavarroPascual,M.M.Neumann,R.Pay ´ a,J.P ´ erez Gonz ´ alez,H.P.Petersson,J.D.Poyato,A.Rochdi,A.Rueda,B.Schreiber, I.P.Shestakov,M.Siles,A.M.Sinclair,A.M.Slinko,R.M.Timoney,A.J.Urena, M.V.Velasco,M.Villegas,A.R.Villena,B.Zalar,andW.Zelazko.
SpecialthanksshouldbegiventoD.Beltita,J.A.Cuenca,H.G.Dales,R.S.Doran, A.Fern ´ andezL ´ opez,A.Kaidi,W.Kaup(RIP),M.Mart´ın,J.F.Mena,J.Mer´ı, A.MorenoGalindo,A.M.Peralta,H.Pfitzner,V.S.Shulman,Yu.V.Turovskii, H.Upmeier,J.D.M.Wright,andD.Yost,fortheirsubstantialcontributionstothe writtingofourwork.
WewouldalsoliketothankthestaffofCambridgeUniversityPressfortheir helpandkindness,andparticularlyR.Astley,C.Dennison,N.YassarArafat, R.Munnelly,N.Saxena,andthecopy-editorK.Eagan.
ThisworkhasbeenpartiallysupportedbytheSpanishgovernmentgrant MTM2016-76327-C3-2-P.
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