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Frontiers in Mathematics Matej Brešar Zero Product Determined Algebras Frontiers in Mathematics AdvisoryEditors WilliamY.C.Chen,NankaiUniversity,Tianjin,China LaurentSaloff-Coste,CornellUniversity,Ithaca,NY,USA IgorShparlinski,TheUniversityofNewSouthWales,Sydney,NSW,Australia WolfgangSprößig,TUBergakademieFreiberg,Freiberg,Germany Thisseriesisdesignedtobearepositoryforup-to-dateresearchresultswhichhavebeen prepared for a wider audience. Graduates and postgraduates as well as scientists will benefit from the latest developmentsat the research frontiers in mathematics and at the “frontiers”betweenmathematicsandotherfieldslikecomputerscience,physics,biology, economics,finance,etc.AllvolumesareonlineavailableatSpringerLink. Moreinformationaboutthisseriesathttp://www.springer.com/series/5388 Matej Brešar Zero Product Determined Algebras MatejBrešar FacultyofMathematicsandPhysics UniversityofLjubljana Ljubljana,Slovenia FacultyofNaturalSciencesandMathematics UniversityofMaribor Maribor,Slovenia ISSN1660-8046 ISSN1660-8054 (electronic) FrontiersinMathematics ISBN978-3-030-80241-7 ISBN978-3-030-80242-4 (eBook) https://doi.org/10.1007/978-3-030-80242-4 Mathematics SubjectClassification: 15A86,16N40,16P10,16R60,16S50,16U40,16W10,16W20,16W25, 17A01,43A20,46H05,46H70,46J10,46L05,47B47,47B48,47B49 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG 2021 Thisworkissubjecttocopyright. Allrightsaresolelyandexclusively licensedbythePublisher,whetherthe wholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storage andretrieval, electronic adaptation, computer software, orbysimilar ordissimilar methodology now knownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublicationdoes notimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbookare believedtobetrueandaccurateatthedateofpublication. Neitherthepublishernortheauthorsortheeditors giveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforanyerrorsoromissions thatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictionalclaimsinpublishedmaps andinstitutionalaffiliations. ThisbookispublishedundertheimprintBirkhäuser,www.birkhauser-science.com,bytheregisteredcompany SpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface An algebra A is said to be zero product determined (zpd for short) if every bilinear functional ϕ on A with the property that ϕ(x,y) = 0 whenever xy = 0 is of the form ϕ(x,y) = τ(xy) for some linear functional τ. If A is a Banach algebra, then we also requirethatϕ andτ arecontinuous. Perhapsthisdefinitiondoesnottellmuch.Theintuitionbehinditisthatmanyproperties of A are determined by pairs of elements whose product is zero. This may still sound vague.Thepoint,however,isthatzpd(Banach)algebrasforma ratherwideclassofnot necessarily associative (Banach) algebras in which problems of different kinds can be solved.Theirtheoryhasreachedacertainlevelofmaturity.Thepurposeofthisshortbook istopresentitinaunifiedandconcisemanner,accessibletoresearchersandstudentsof differentbackgrounds.Thematerialismostlytakenfromnumerouspaperspublishedover thelast15years,butsomenewresultsandnewproofsofknownresultsarealsoincluded. Theconceptof a zpdalgebraarosefromtwo unrelatedpapers,[53] and[4]. Thefirst one,writtenjointlywithP.Šemrl,studiescommutativitypreservinglinearmapsoncentral simple algebras,andthe secondone,written jointlywith J. Alaminos,J. Extremera,and ∗ A. Villena, studies local derivations and related maps on C -algebras. Their common feature is that the results on certain linear maps were derived from the consideration of bilinear maps that vanish on pairs of elements whose product is zero. This eventually led to the introduction of the concept of a zpd algebra and its systematic study, which evolved in two parallel directions: the algebraic and the (in many ways richer) analytic. Someterminologicalmisfortunesoccurredontheway:theanalyticbranchmostlyusesthe term“BanachalgebrawithpropertyB”(introducedin[6],inconjunctionwiththerelated propertyA)fortheconceptthatissimilartothatofa“zeroproductdeterminedalgebra” usedin the algebraicbranch.Oneofthe aimsofthebookisto unifybothbranches,and alongtheway we willmakesometerminologicaladjustments.PropertyB willtherefore playonlyanauxiliaryrole. The book is divided into three parts. Part I considers the algebraic theory and Part II theanalytictheory.PartIIIisdevotedtoapplications,thatis,itexaminesproblemsfrom differentareasofmathematicsforwhichthezpdconcepthasproveduseful.Someresults in this last part are stated without detailed proofs. This is because the topics treated are v vi Preface very diverse and so many of them would need their own technical introduction. In the courseofwritingIrealizedthatitisbettertorefertooriginalsourcesfordetailsandfocus primarilyonideasthatillustratetheusefulnessoftheconceptsstudiedinthisbook. Parts of the book present results obtained jointly with my colleagues Jeronimo Alaminos,JoseExtremera,andArmandoVillenafromtheUniversityofGranada.Iwould like to thank them for the fruitful and pleasant long-term collaboration (and for being wonderfulhostsduringmyfrequent,enjoyablevisitstoGranada).Theideaforthebook arose from discussions with Armando, who was the leading force in developing the analytic branch of the theory. Although he did not join me as a coauthor, his influence canbefeltashehasbeengivingmecontinuoushelpintheprocessofwriting.Myspecial thankstohim! Finally,IwouldalsoliketothankŽanBajukforpointingoutsomeerrorsinanearlier version,andtherefereesforusefulcomments. LjubljanaandMaribor,Slovenia MatejBrešar April2021 Contents PartI AlgebraicTheory 1 ZeroProductDeterminedNonassociativeAlgebras ............................ 3 1.1 TheDefinitionofazpdNonassociativeAlgebra............................. 3 1.2 SymmetricallyzpdNonassociativeAlgebras................................. 10 1.3 StabilityUnderAlgebraicConstructions ..................................... 11 2 ZeroProductDeterminedRingsandAlgebras.................................. 17 2.1 zpdRings....................................................................... 17 2.2 ExamplesandNon-examplesofzpdAlgebras ............................... 20 2.3 TheFinite-DimensionalCase.................................................. 28 3 ZeroLie/JordanProductDeterminedAlgebras................................. 33 3.1 LieAlgebrasandJordanAlgebras ............................................ 33 3.2 zLpdAlgebras.................................................................. 36 3.3 zJpdAlgebras .................................................................. 48 PartII AnalyticTheory 4 ZeroProductDeterminedNonassociativeBanachAlgebras................... 61 4.1 CharactersandtheLimitationsoftheAlgebraicApproach.................. 61 4.2 TheDefinitionofazpdNonassociativeBanachAlgebra.................... 65 4.3 PointDerivations............................................................... 71 5 ZeroProductDeterminedBanachAlgebras..................................... 75 5.1 PropertyB...................................................................... 75 5.2 StabilityUnderAnalyticConstructions....................................... 79 5.3 ExamplesandNon-examplesofzpdBanachAlgebras ...................... 84 vii viii Contents 6 ZeroLie/JordanProductDeterminedBanachAlgebras....................... 95 6.1 TheConditionxy =yx =0 .................................................. 95 6.2 zLpdBanachAlgebras......................................................... 101 6.3 zJpdBanachAlgebras ......................................................... 104 PartIII Applications 7 HomomorphismsandRelatedMaps.............................................. 113 7.1 ZeroProductPreservingMaps................................................ 113 7.2 CommutativityPreservingMaps.............................................. 121 7.3 JordanHomomorphisms....................................................... 126 8 DerivationsandRelatedMaps .................................................... 139 8.1 CharacterizingDerivationsbyActiononZeroProducts..................... 139 8.2 LocalDerivations .............................................................. 145 8.3 DerivationsandQuasinilpotentElements..................................... 151 9 Miscellany............................................................................ 157 9.1 CommutatorsandSpecial-TypeElements.................................... 157 9.2 OrthogonalityConditionsonn-LinearMaps................................. 162 9.3 NonassociativeProductsofMatrices.......................................... 168 References................................................................................ 175 Index...................................................................................... 183 Part I Algebraic Theory