Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles: Basic Representation Theory of Groups and Algebras

753 Pages·1988·10.9 MB·English

by  J.M.G. Fell and R.S. Doran (Eds.)

Representations of *-Algebras,
Locally Compact Groups, and
Banach *-Algebraic Bundles
Volume 1
Basic Representation Theory of
Groups and Algebras
J. M.G.Fell
Department ofMathematics
University ofPennsylvania
Philadelphia, Pennsylvania
R. S. Doran
Department ofMathematics
Texas Christian University
Fort Worth, Texas
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Library ofCongressCataloging-in-Publication Data
Fell,1.M.G.(James MichaelGardner), Date
Representationsof*-algebras,locallycompact
groups, and Banach *-algebraicbundles.
(Pureand applied mathematics; 125-126)
Includes bibliographies and indexes.
Contents: v,I. Basicrepresentation theory of
groups and algebras - v,2. Banach*-algebraic
bundles,induced representations,and thegeneralized
Mackey analysis.
I. Representations ofalgebras. 2. Banach algebras.
3. Locallycompact groups. 4. Fiber bundles
(Mathematics) I. Doran,Robert S.,Date
II. Title. III. Series:Pure and applied mathematics
(Academic Press); 125-126.
QA3.P8 vol.125-126[QA326] 510s[512'.55] 86-30222
ISBN0-12-252721-6(v, I:alk,paper)
ISBN0-12-252722-4(v.2:alk.paper)
88 89 90 91 9 8 765 4 3 2 I
Transferred to Digital Printing 2005
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Preface
This two-volume work serves a double purpose, roughly (but not entirely)
corresponding to the division into volumes. The first purpose (which
occupies Volume 1and part ofVolume 2)isto provide a leisurelyintroduc
tion to the basic functional analysis underlying the theory of infinite
dimensional unitary representations of locally compact groups and its
generalization to representations ofBanach algebras. Thus, the first volume
covers generally well-known material on measure theory (Chapter II),
topological groups and their group algebras (Chapter III), purely alge
braic representation theory (Chapter IV), Banach algebras (Chapter V),
C*-algebras and their *-representations (Chapter VI), and the topology of
*-representations (Chapter VII). Chapters IX and X (in Volume 2), which
deal mostly with classical material on compact groups and locally compact
Abelian groups respectively,also serve primarily the first purpose.
Thesecond purpose(whichoccupiesChaptersVIII,XI,and XIIofVolume
2) is to bring the reader to the frontiers of present-day knowledge in one
limited but important area, namely, the Mackey normal subgroup analysis
and its generalization to the context of Banach *-algebraic bundles. Here
too our presentation is quite leisurely. Chapter VIII deals with Banach
*-algebraic bundles and their representations. These objects can be thought
ofroughly as Banach *-algebras whichare"over"abasegroup, and inwhich
xiii
xiv Preface
multiplication and involution are "covariant" with the operations of multi
plicationand inverseinthe basegroup. The primarymotivatingexampleofa
Banach *-algebraic bundle isin fact (very roughly speaking)just the group
algebraofa locally compactgroup; the more general objects in thiscategory
can be regarded as "generalized group algebras." Chapters XI and XII are
devoted to the "bundle generalization" (i.e., the generalization to Banach
*-algebraicbundles)ofthe Mackey normalsubgroupanalysis. In itsclassical
form developed by Frobeniusfor finitegroups and by Mackeyfor separable
locally compact groups, this is a powerful technique for classifying the
unitary representations of a group H when we are given a closed normal
subgroup N of H; under appropriate conditions it classifiesthe irreducible
representations ofH interms ofthose ofN and ofsubgroupsofthe quotient
groupG= H/N. More generally,supposewearegivenaBanach*-algebraA,
together with some suitable Banach *-algebraic bundle structure 91 for A
over some locally compact group G. The "bundle generalization" of the
normalsubgroupanalysis willthen enableustoclassifythe *-representations
ofAinterms ofthose ofsubgroupsofGand those oftheso-called unit fiberof
fA (corresponding in the classical case to the normal subgroup N). The
fundamental tools oftheclassicalnormalsubgroupanalysis are theconstruc
tion ofinduced representationsand the ImprimitivityTheorem.These willbe
developedinChapterXIinthe"bundlecontext"(infactinanevenmoregen
eral, purely algebraic form). Finally, in Chapter XII we take up the Banach
*-algebraicbundleversionoftheMackeynormalsubgroupanalysis itselfand
discuss numerousexamples. (We remark that weusethe phrase"generalized
normalsubgroupanalysis"or"generalized Mackey analysis:'eventhough in
thegeneralized bundle version nonormalsubgroupsofagroupneedappear!)
The choice of the material presented in Volume 1is of course to a large
extent governed by the needs of Volume 2.For example, little issaid about
von Neumann algebras and nothing about non-commutative direct integral
decomposition theory, since it turns out (as mentioned laterin this Preface)
thatthesetopics are hardlyneeded inthe Loomis-Blattner-Glimmversion of
the normalsubgroupanalysis.Onthe otherhand,inChapterVII wego quite
deeply into the topology of the space of all *-representations of a given
*-algebra.
It is presumed that the reader has some prior knowledge of elementary
group and ring theory, general topology, general linear topological spaces,
and abstract measure theory (though a rapid introduction to the latter is
given in Chapter II, from aslightly novel standpointand much ofit without
proofs). With this equipment the reader should have no difficultyin reading
the present work.
Preface xv
Aword about the historical backgroundand genesisofthis work might be
ofinterest here.
The story goes back to the summerof 1955,when the first-named author
attendedthesummercourse on grouprepresentationsgivenat the University
ofChicago byG. W. Mackey. The outcomeofthatcourse wasaset ofnotes,
prepared by him jointly with the late D. B. Lowdenslager, presenting the
three stages of the program developed by Mackey in the course for the
classification ofseparablegrouprepresentations,namely: 1)Global multipli
city theory (the theory of von Neumann algebras); 2) direct integral
decomposition theory; and 3) the normal subgroup analysis based on
Mackey's version of the Imprimitivity Theorem. These notes made no
pretenseofgivingcompleteproofs ofthe major results, because ofthe wealth
oftechnical preliminaries(mostly measure-theoretic)which would have been
necessaryinorderto do so.However, thefirst-namedauthorhad bythis time
becomefascinated with the wholesubject ofsymmetryand grouprepresenta
tions and made up his mind to write an exhaustive account of the subject
sooner or later. The present work is the outcome of that intention.
However, asevenanoddingacquaintance with theearliernotes willshow,
the plan ofthose notes wasquite differentfromthe presentwork. Both aim at
the normal subgroup analysis as their climax, but there the similarity ends.
Two mathematical discoveries in the intervening years account for much of
the difference of approach. First, a new approach to the theory of induced
representations, the ImprimitivityTheoremand the normal subgroupanaly
sisfor groups wasdeveloped by L.H. Loomis and R.J.Blattnerin the early
1960s,which dispenses entirely with Mackey's separability assumptions and
substantially reduces the role of measure theory. A key factor in the
Loomis-Blattnerdevelopment wasan observation byJ.G. Glimm (in 1962)
to theeffectthatjustthatportionofdirect integral theorywhichisneeded for
the Mackey normal subgroup analysis can be developed without any
separabilityassumptionswhatever. In viewofthesediscoveries wedecided to
orient the present work toward the Imprimitivity Theorem and the normal
subgroup analysis in the non-separablecontext, using the results of Loomis,
Blattner,and Glimm ratherthan Mackey'soriginalhighlymeasure-theoretic
approach.
This is not to say that the Loomis-Blattner-Glimm approach supersedes
that of Mackey. Indeed, Mackey's measure-theoretic proofofthe Irnprimiti
vity Theorem gives useful information in the separable context which does
not emergeinthe Loomis-Blattnerproof.Also,hisdirect integraldecomposi
tion theory gives a beautiful insight into the structure of arbitrary unitary
representations ofseparable Type I groups which isquite unavailable in the
xvi Preface
non-separablecontext that wehave chosen to work with inChapters Xl and
XU.
The second discovery that has molded the presentwork isthe realization,
hinted at earlier in this Preface, that the natural context for the normal
subgroup analysis consists not merely in locally compact group extensions,
but in the more general structures that wecallsaturated Banach ·-algebraic
bundles over locally compactgroups. The firststep in this direction (at least
for infinite-dimensional representations) seems to have been taken by M.
Takesaki, who in 1967extended the normal subgroup analysis to crossed
products,or semidirectproducts,ofBanach ·-algebrasand groups. Notlong
after (in 1969) the first-named present author showed that the normal
subgroup analysis is valid for the wider class of so-called homogeneous
Banach·-algebraicbundles.(Oneshouldalso mentionthework ofH.Leptin,
who in 1972demonstrated the same for his generalized fill algebras, which
are very closely related to homogeneous Banach ··algebraic bundles.)
Finally, it became clear that the normal subgroupanalysis actually requires
only the property of saturation of the Banach ··algebraic bundle, which is
considerably weaker than homogeneity. We therefore decided to present the
whole development of Chapter XU (the culminating point of the present
work) in the context ofmerely saturated Banach ·-algebraic bundles.
In hisSpringer Lecture Notes Volume of 1977(Fell [17]), the first-named
author proved the Irnprimitivity Theorem for saturated Banach ·-algebraic
bundles, but the further step ofdeveloping the generalized normalsubgroup
analysis in that context was not taken up there. Chapter XII ofthepresent
work is therefore thefirst exposition ofthe Mackey normal subgroupanalysis
for saturated Banach ·-algebraic bundles whichhasappeared inprint. (How
ever,it must bepointedout thatourmethodsare basically the sameas those
of Blattner in his 1965paper on the normal subgroup analysis for the non
separablegroupcase.No realadvanceintechniqueisneeded inorderto pass
to the more general situation.)
Actually, the Irnprirnitivity Theorem for Banach ·-algebraic bundles will
be obtained in Chapter XI as (more or less)a corollary of a more general,
purely algebraicIrnprimitivityTheoremthatmakes no referenceto groupsat
all. Not only does this generalized Imprimitivity Theorem have applications
inareas outsideofthe theoryofgroups,but itsuggests the hope ofeventually
"algebraizing" the entire Mackey normal subgroup analysis, i.e.,of present
ing it in the context of algebras, and banishing groups from their present
crucial role as base spaces ofthe Banach ·-algebraic bundles. Realization of
this hope would be an important advance in the program of"algebraizing"
all of the theory ofgroup representations-a program begun in the earliest
Preface xvii
stagesofthe workofGelfand and Naimarkon Banach *-algebras,whenthey
showed that thefactofexistenceof"enough"irreducible unitaryrepresenta
tions oflocallycompactgroups wasonlya specialcaseofthecorresponding
factfor Banach *-algebras. However, the hope ofcompletely "algebraizing"
the normal subgroup analysis has not yet been fulfilled-a disappointment
that ispartly responsibleforthedelayintheappearanceofthepresent work.
Although we have taken pains to present the whole theory of induced
representations,theImprimitivityTheorem, and thenormalsubgroupanaly
sisinthegeneralizedcontext ofBanach *-algebraicbundles,thereisnodoubt
that the "group case" (where we are interested in classifying the
unitary representations of a group that is presented to us as a group
extension) is the case of greatest interest. We have therefore taken every
occasiontopoint outexplicitlyhowour resultsspecializetothe"groupcase."
We hope thereby to minimize the annoyance feltby those (and wesuspect
theyarequiteafew!)whoseaimistolearnfromtheworkonlysomuchofthe
subject as isrelevant to groups.
Letusnowsayafewwordsabouttheexercisesthat havebeenplacedat the
endsofthevariouschapters. Someoftheseinvolvecheckingroutinedetailsof
proofsorremarks that havebeenlefttothereader.Othersinvolvenewresults
and actual extensions of the theory, while still others provide examples or
counter-examples to various assertions in the text. The exercises range in
difficultyfrom extremely easy to rather difficult.Some of them have been
provided with hints for their solution.
Throughout both volumesofthiswork the reader willobservethat certain
propositions and results have been starred with an asterisk. Generally
speaking, thesepropositionscontaininformationand insights whichmay be
ofinterest to the reader but which are not essential to the main flowofthe
material being discussed. Their proofs have therefore either been omitted
entirely or else only sketched. As part of the exercisesthe reader has been
asked to supply proofs (or to fillin details ofproofs which have only been
sketched) to many of thesestarred results.
Anearlier draft ofthiswork had beenmore or lesscompleted by the first
named authorin 1975(the present second-namedauthorwasnot involvedin
this effort except in a peripheral way with proof-reading and offering
suggestions here and there). Because of various difficulties,the full manu
script was not published at that time (although in 1977 the material of
ChapterXIand the first part ofChapterVIII waspublished as the Springer
Lecture Notes Volume referred to above).
Duringthe spring of1981,thesecond-namedauthorspent thesemesteras
a Member of The Institute for Advanced Study. The close proximity of
xviii Preface
Princeton to Philadelphia made it possible for the two of us to easily get
together. It was decided at that time that the whole manuscript should be
published, and, starting in 1982, the second-named author, assisted by the
first-named, undertooka revisionand updatingoftheentiremanuscript.The
present two-volume work is the result of these efforts.
This Prefacewouldnotbecompletewithouta warmacknowledgement,on
the part ofthe first-named author,ofthe debt which he owes to ProfessorG.
W. Mackey for introducing him to the subject of group representations and
guiding his subsequent study of it. Professor Mackey's dedication to and
enthusiasm for the subject is contagious, and the first-named author's
mathematicallifewas permanentlyshifted into newchannels by hisexposure
to Mackey's 1955summer course in Chicago and subsequent contacts with
him over the years.
In closing, we want to thank the National Science Foundation for its
financialsupportofthe first-named author'ssabbaticalleave during 1970-71,
when the original manuscript began to take shape. The first-named author
also wishes to express his appreciation ofthe hospitalityextended to him by
the NaturalSciences InstituteofTheUniversityofIceland,Reykjavik,during
hissabbaticalleave thereinthe fallof 1985-aperiodspentlargelyinputting
final touches to this manuscript. Special thanks are due to Professor Marc
Rieffelfor the many conversations with him in the early 1970s,which helped
to giveChapterXI its present form. Wealso wish to thank Professor Robert
C. Busby for carefully reading an early draft ofChapters II through VII and
offering a numberofuseful suggestions. We also express our warm apprecia
tion of the excellent work of Paula Rose in typing the entire first manuscript
(during the early 1970s). Likewise our special thanks are due to Shirley
Doran, who has carried through the endless revisions and additions the
manuscript has undergone in the past several years, and to the Texas
Christian University Department of Mathematics for making its office
facilities and resources available to usduring this period.
We want also to express our gratitude to pur wives Daphne and Shirley
(and our children also) for providing the encouragement and happy home
environment that have added so much to the purely mathematical joys of
seeing this long-delayed project to completion. But, most of all, we want to
express,at leastinsometiny measure,ourgratitudeto ourCreatorand God,
who has given us the strength, the education, and the opportunity both to
enjoy and to share with others the beauties of mathematical discovery.
J. M. G. Fell
R. S. Doran
There are things whichseemincredible to
most men who havenot studied mathematics.
- Archimedes
Introduction to Volume 1
(Chapters I to VII)
This Introduction isdesigned to orient the reader toward the subject-matter
not only of Volume 1but of Chapters IX and X of Volume 2 as well.The
reader who isnewto the subject willfindpartsofitquite hard goingat first.
However,itistobehoped that,intheprocessofassimilatingthedetailsofthe
subject, hewilloccasionally return to this Introduction and findit helpfulin
gaining a usefuloverviewofthe field.
Thestrongestmotivationforthestudy ofinfinite-dimensionalalgebras and
their representations comes from the representation theory of locally com
pact groups. For this reason, although the greater part of Volume 1of this
work deals withthe representationtheory ofalgebras, most ofthis Introduc
tion will be spent in motivating the theory of group representations, from
whichmany oftheproblems and achievements ofthe more general theory of
representations ofalgebras took their origin.
The theory of group representations has its historical roots in two
developments ofnineteenth-centurymathematics,bothofthe utmostimpor
tance in the history ofmathematics and ofmathematical physicsas well.
The first of these is the theory of Fourier series, whose importance for
appliedmathematicsstemsfromitsapplication toproblems ofvibrationand
1