Jordan Structures in Geometry and Analysis

272 Pages·2012·1.217 MB·English

by  Cho-Ho Chu

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Jordan Structures in Geometry
and Analysis
CHO-HO CHU
QueenMary,UniversityofLondon
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,Sa˜oPaulo,Delhi,Tokyo,MexicoCity
CambridgeUniversityPress
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Contents
Preface pageix
1 JordanandLietheory 1
1.1 Jordanalgebras 1
1.2 Jordantriplesystems 13
1.3 LiealgebrasandtheTits–Kantor–Koecherconstruction 39
1.4 MatrixLiegroups 50
Notes 61
2 Jordanstructuresingeometry 62
2.1 BanachmanifoldsandLiegroups 62
2.2 Riemannianmanifolds 82
2.3 JordanalgebrasandRiemanniansymmetricspaces 101
2.4 JordantriplesandRiemanniansymmetricspaces 121
2.5 Jordantriplesandsymmetricdomains 144
Notes 169
3 Jordanstructuresinanalysis 171
3.1 Banachspaces 171
3.2 Holomorphicmappings 183
3.3 ContractiveprojectionsonJB*-triples 202
3.4 IsometriesbetweenJB-triples 217
3.5 Hilbertspaces 227
Notes 248
Bibliography 251
Index 257
vii

Preface
Despite the rapid advances in Jordan theory and its diverse applications in
the last two decades, there are few convenient references in book form for
beginnersandresearchersinthefield.Thisbookisamodestattempttofillpart
ofthisgap.
Theaimofthebookistointroducetoawidereadership,includingresearch
students,thecloseconnectionsbetweenJordanalgebras,geometry,andanaly-
sis.Inparticular,wegiveaself-containedandsystematicexpositionofaJordan
algebraicapproachtosymmetricmanifoldswhichmaybeinfinite-dimensional,
and some fundamental results of Jordan theory in complex and functional
analysis.Inshort,thisbookisaboutJordangeometricanalysis.
AlthoughtheconceptofaJordanalgebrawasintroducedoriginallyforquan-
tumformalism,byP.Jordan,J.vonNeumannandE.Wigner[64],unexpected
and fruitful connections with Lie algebras, geometry and analysis were soon
discovered.Inthelastthreedecades,manymoreapplicationsofJordanalge-
braicstructureshavebeenfound.Weexposesomeoftheseapplicationsinthis
book.Needlesstosay,thechoiceoftopicsisinfluencedbytheauthor’spredilec-
tions, and regrettable omissions are inevitable if the length of the book is to
bekeptmanageable.Nevertheless,anefforthasbeenmadetocoversufficient
basicresultsandJordantechniquestoprovideahandyreference.
We begin by discussing the basic structures of Jordan algebras and Jordan
triplesystemsinChapter1,andtheconnectionsoftheseJordanstructuresto
Lietheory.AnimportantlinkistheTits–Kantor–Koecherconstruction,which
establishes the correspondence between Jordan triple systems and a class of
gradedLiealgebras.WediscusssomedetailsofclassicalmatrixLiegroupsand
theirLiealgebrasandusethemasexamplestoillustratetheseconnections,as
wellaspreparationfortheintroductionofBanachLiegroupsinthefollowing
chapter.
ix