Table Of ContentFrontiers in Mathematics
Matej Brešar
Zero
Product
Determined
Algebras
Frontiers in Mathematics
AdvisoryEditors
WilliamY.C.Chen,NankaiUniversity,Tianjin,China
LaurentSaloff-Coste,CornellUniversity,Ithaca,NY,USA
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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
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2021
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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
