Non-commutative Gelfand theories: A tool-kit for operator theorists and numerical analysts

398 Pages·2011·2.5 MB·English

by  Steffen Roch

Universitext
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(cid:2) (cid:2)
Steffen Roch Pedro A. Santos
Bernd Silbermann
Non-commutative
Gelfand Theories
A Tool-kit for Operator Theorists
and Numerical Analysts
SteffenRoch Prof.BerndSilbermann
TechnischeUniversitätDarmstadt TechnischeUniversitätChemnitz
FachbereichMathematik FakultätfürMathematik
Schloßgartenstraße7 ReichenhainerStraße39
64289Darmstadt 09126Chemnitz
Germany Germany
[e-mail] [e-mail]
PedroA.Santos
UniversidadeTécnicadeLisboa
InstitutoSuperiorTécnico
DepartamentodeMatemática
Av.RoviscoPais,1
1049-001Lisboa
Portugal
[e-mail]
Editorialboard:
SheldonAxler,SanFranciscoStateUniversity
VincenzoCapasso,UniversitàdegliStudidiMilano
CarlesCasacuberta,UniversitatdeBarcelona
AngusJ.Macintyre,QueenMary,UniversityofLondon
KennethRibet,UniversityofCalifornia,Berkeley
ClaudeSabbah,CNRS,ÉcolePolytechnique
EndreSüli,UniversityofOxford
WojborWoyczynski,CaseWesternReserveUniversity
ISBN978-0-85729-182-0 e-ISBN978-0-85729-183-7
DOI10.1007/978-0-85729-183-7
SpringerLondonDordrechtHeidelbergNewYork
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Preface
Thecentralnotioninthisbookisthatofalocalprinciple.Localprinciplesprovide
an abstract frame for the natural and extremely useful idea of localization, i.e. to
divide a global problem into a family of local problems. The local principles the
readerwillencounterinthistextareformulatedinthelanguageofBanachalgebras
andcanbecharacterizedasnon-commutativeGelfandtheories.Theynowforman
∗
integral part of the theory of Banach andC -algebras, and they provide an indis-
pensabletooltostudyconcreteproblemsinoperatortheoryandnumericalanalysis.
More than thirty years ago, Douglas derived the Gohberg-Krupnik symbol cal-
culus for singular integral operators via a combination of a local principle (which
nowbearshisname)andHalmos’twoprojectionstheorem.Aroundthesametime
Kozakprovedtheequivalencebetweenthestabilityofanoperatorsequenceandthe
invertibilityofarelatedelementinacertainBanachalgebrawhichhethenstudied
bySimonenko’slocalprinciple.Sincethattimetherehaveappeareddozensofpa-
perswheretheideaoflocalizationhasbeenused,furtherdeveloped,andappliedin
severalcontexts.Astheoutcomeofthisdevelopment,wenowhaveapowerful,rich
and beautiful theory of algebraic localization, the principles and results of which
arewidelyscatteredintheliterature.Thelackofageneralcontext,andtheuseof
different notation from paper to paper make it difficul for the researcher and the
graduate student to familiarize themselves with the theory behind local principles
andtomakeuseoftheseresultstostudytheirownproblems.
It is this defect that the present book seeks to solve. It started as a much sim-
pler task: an updated re-edition of an out-of-print report [168], back in 1998. The
changingobjectivesandprofessional obligationsof theauthors kepton increasing
the scopeanddelayingthework. After morethanten years, we are finall able to
presentit.Wethinkthatthedelayhasbeenworthit,andthereaderhasareadable
andusefultextinhishands.
It is our intention that this work be a basic but complete introduction to local
principles, formulated in the language of Banach algebras, that allows the reader
to get a general view of the area and enables him to read more specialized works.
Many results that appeared in periodicals or reports, and can be hard to find are
presented,streamlinedandcontextualizedhere,andtherelationswithotherresults
vii
viii Preface
are made clear. Some results which were in complete form available only in the
Russianliterature,likethelocalprinciplebySimonenko,areincluded.Andfinall ,
afewexistinggapsinthetheoryarefille inwithfullproofs,whichappearherefor
thefrsttime.
ThetextstartswithachapterontherelevantnotionsforlocalprinciplesofBa-
nachalgebratheory.Assuch,thefrstpartcanserveasatextbookforaonesemester
graduate course on Banach algebras with emphasis on local principles. Exercises
andexamplesaregiventhroughoutthetext.Wefocusonapplicationstosingularin-
tegraloperatorsandconvolutiontypeoperatorsonweightedLebesguespaces.The
choiceofapplicationsistheresultbothofourparticularinterestsasresearchersand
of the genetic inheritance of the text, which was born as a report on algebras of
convolutiontypeoperators.
Mostfigure inthebookwereproducedwiththehelpofMathematica1.Acouple
of figure were produced with Adobe Illustrator2. The authors acknowledge the
research center CMA and its successor CEAF (Portugal) for travel and meeting
supportduringthewritingofthebook.
WewouldliketothankourcolleaguesMarkoLindnerandHelenaMascarenhas
for their stimulating and helpful discussions during the work on this text. We are
speciallygratefultoAlexeiKarlovich,whocarefullyreadthemanuscriptandgave
manyvaluablesuggestionsforitsimprovement.
OursincerethanksgoestoSpringerandKarenBorthwickforincludingthebook
intheUniversitextseriesandforpleasantandhelpfulco-operation.
Chemnitz,Darmstadt,Lisboa SteffenRoch
December2009 PedroA.Santos
BerndSilbermann
1MathematicaisaregisteredtrademarkofWolframResearch,Inc.
2AdobeIllustratorisatrademarkofAdobe,Inc.
Contents
Preface............................................................ vii
Introduction ....................................................... xi
PartI Non-commutativeGelfandTheories
1 Banachalgebras ............................................... 3
1.1 Basicdefinition ........................................... 3
1.2 Invertibilityandspectrum.................................... 17
1.3 Maximalidealsandrepresentations ........................... 35
1.4 SomeexamplesofBanachalgebras ........................... 45
1.5 Notesandcomments........................................ 60
2 Localprinciples................................................ 63
2.1 Gelfandtheory............................................. 64
2.2 Allan’slocalprinciple....................................... 72
2.3 Norm-preservinglocalization................................. 85
2.4 Gohberg-Krupnik’slocalprinciple ............................ 91
2.5 Simonenko’slocalprinciple.................................. 94
2.6 PI-algebrasandQI-algebras..................................106
2.7 Notesandcomments........................................116
3 Banachalgebrasgeneratedbyidempotents .......................121
3.1 Algebrasgeneratedbytwoidempotents........................121
3.2 AnN idempotentstheorem ..................................129
3.3 Algebraswithfli changingtheorientation .....................150
3.4 Algebraswithfli preservingtheorientation ....................165
3.5 Coefficien algebras.........................................172
3.6 Notesandcomments........................................186
ix