Operator algebras and quantum statistical mechanics 1 2nd edition

At the time ofwriting the first edition it was an open question whethermaximal orthogonal probability measures on the state space of a C*-algebra were automatically maximal among all the

Texts and Monographs in Physics

Series Editors:

R Balian, Gif-sur-Yvette,France

W Bei9lb6ck, Heidelberg,t, Germany

H.Grosse, Wien,Austria

E H.Lieb, Princeton, NJ,USA

N.Reshetikhin,Berkeley,CA,USA

H Spohn,Minchen,Germany

W.Thirring,Wien,Austria

Home page:http:Hwww.math uio no/-bratteli/

ProfessorDerek W.Robinson

Australian NationalUniversity

School of Mathematical Sciences

ACT 0200Canberra,Australia

e-mail: Derek Robinson@ anu edu au

Home page:http:Hwwwmaths.anu.edu.au/-derek/

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Bratteli,Ola. Operator algebrasand quantum statistical mechanics (Texts andmonographsinphysics) Bibliography;v

1, p. Includes index Contents: v 1 C *

ISBN3-540-17093-6 2nd EditionSpringer-VerlagBerlinHeidelbergNew York

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Preface to the Second Printing of the Second Edition

Inthis secondprintingof the second edition several minor andonemajormatical mistake have been corrected Weare indebtedtoRoberto Conti, SindreDuedahl and Reinhard Schaflitzel forpointingtheseout.

DerekW.Robinson

Preface to the Second Edition

Thesecond edition of this book differs from theoriginal in threerespects First,

we have eliminated a large number of typographical errors. Second, we havecorrected a small number of mathematical oversights. Third, wehaverewrittenseveral subsectionsinordertoincorporatenew orimprovedresults.Theprincipal changesoccurinChapters3 and 4

In Chapter 3, Section 3.1.2 now contains a more comprehensive discussion

ofdissipative operators andanalytic elements Additions andchanges have alsobeen made in Sections 3.1.3, 3.1.4, and 3.1.5 Further improvements occurinSection 3.2.4.InChapter4theonly substantialchangesare toSections 4.2.1 and4.2.2 At the time ofwriting the first edition it was an open question whethermaximal orthogonal probability measures on the state space of a C*-algebra

were automatically maximal among all the probability measures on the space.This question was resolved positively in 1979 and the rewritten sections now

incorporatetheresult

All these changes are nevertheless revisionary in nature and do not change

the scope of the original edition Inparticular, wehave resisted the temptation

todescribe thedevelopments of the lastsevenyears in thetheory ofderivations,

anddissipations,associated withC*-dynamicalsystems.Thecurrent stateof this

theoryis summarized in[[Bra1]]publishedinSpringer-Verlag'sLectureNotes inMathematics series

DerekW.Robinson

v

Preface to the First Edition

In this book we describe the elementary theory ofoperator algebras andparts of the advanced theory which are of relevance, or potentially of

relevance, to mathematical physics Subsequently we describe variousapplicationsto quantum statistical mechanics At theoutset of thisproject

weintendedto coverthis material inonevolume but in thecourseof

develop-mentitwasrealized that this would entail the omission of variousinteresting

topics or details Consequently the book was split into two volumes, thefirst devotedtothegeneral theoryofoperatoralgebrasand the secondtotheapplications.

This splittingintotheoryand applicationsis conventional but somewhatarbitrary. In the last 15-20 years mathematicalphysicists have realized theimportance ofoperator algebras and their states and automorphisms forproblemsof fieldtheoryand statistical mechanics But thetheoryof 20 yearsago was largely developed for the analysis of grouprepresentations and it

was inadequate for many physical applications. Thus after a short

honey-moonperiodin which thenewfound tools of theextanttheorywereapplied

tothemostamenableproblemsalongerandmoreinteresting periodensued

in which mathematical physicists were forced to redevelop the theory inrelevant directions Newconceptswereintroduced,e.g.asymptoticabelian-

ness and KMS states, new techniques applied, e.g the Choquet theory ofbarycentric decompositionfor states, and new structural results obtained,

e.g the existence ofacontinuum ofnonisomorphic type-threefactors Theresults of thisperiodhadasubstantialimpactonthesubsequent development

of thetheoryofoperatoralgebrasand ledto acontinuing periodof fruitful

Vil

collaboration between mathematicians and physicists They also led to an

forced the formation of thetheory. Thus in this contextthe division of thisbook hasacertainarbitrariness

Thetwovolumes of the book contain sixchapters,four in this first volumeand two in the second The chapters of the second volume are numberedconsecutively with those of the first and the references are cumulative.Chapter I is a brief historical introduction and it is the five subsequent chaptersthat form the mainbodyof material We have encountered variousdifficulties inourattemptstosynthesizethis material intoonecoherent book.Firstlytherearebroad variations in thenatureanddifficultyof the differentchapters. This is partly because the subject matter lies between the main-streamsof pure mathematics and theoreticalphysicsandpartlybecause it isa

mixture of standard theory and research work which has not previously appearedin book form We have triedtointroduceauniformityandstructureand we hopethe reader will find ourattemptsare successful Secondlytherange oftopics relevant to quantum statistical mechanics is certainlymore

extensive than our coverage. For example we have completely omitteddiscussion of open systems, irreversibility, and semi-groups ofcompletely positivemaps because thesetopics have been treated in otherrecent mono-

graphs [[Dav 1]] [[Eva 1]].

This book was written between September 1976 and July t979.Most ofChapters 1-5 were writtenwhilst the authors were in Marseille at

the Universit6 d'Aix-Marseille 11, Luminy, and the Centre de Physique Th6oriqueCNRS During asubstantial part of this period0 Bratteli was

supported bytheNorwegianResearch Council for Science and Humanitiesandduringthecomplementary period byapostof"

pleasure to thank Mlle Maryse Cohen-Solal, Mme Dolly Roche, andMrs.MaydaShahinianfor their work

We haveprofitedfrom discussions with many colleagues throughoutthepreparationof the manuscript.WearegratefultoGavin Brown,Ed Effros,

George Elliott, Uffe Haagerup, Richard Herman, Daniel Kastler, Akitaka

Kishimoto, John Roberts, Ray Streater and Andr6 Verbeure for helpful

commentsand correctionsto earlier versions

We are particularly indebted to Adam Majewski for reading the finalmanuscriptandlocatingnumerous errors.

Derek W Robinson

Contents (Volume 1)

Introduction

2.2.3 ApproximateIdentities andQuotient Algebras 39

2.4 vonNeumannAlgebras 65

2.4.2 Definition andElementaryPropertiesofvonNeumannAlgebras 71

2.5 Tomita-Takesaki ModularTheoryand Standard Forms ofvonNeumann

2.5.3 IntegrationandAnalyticElements for One-ParameterGroupsof

2.7.3 Weights on Operator Algebras; Self-Dual Cones of General

vonNeumannAlgebras; Dualityand Classification ofFactors;

(d) ifWpossessesanidentity,A !! B >0,and A> 0 then

PROOF (a) WeadjoinanidentityI toW if necessary Thespectralradius formula

of Theorem 2.2.5 then givesA < 11A111 and hence 0:!! B :!! 11A11T But thisimplies

thatJIB 11 :! 11 All byasecondapplicationof thesameformula

(b) One hasa(A - 11A111/2) g [-11A11 /2, 11A11 /2] and henceu((A - 11A111/2)2)

[0, IIA112/4] byTheorem2.2.5(d).Thus

(B+ AT)-1/2

(A + AT)(B + AT)-1/2 >I

If, however, X =X* and X >- T then u(X)g [1, oo> and u(X-')- [0, 1] by

Proposition 2.2.3 ThusX-' -< 1.Thisgives

B1/2

C*-Algebras Algebras

i.e., A1/2 > B1/2 > 0 By use of similar transforms one can deal with otherfractional powers and deduce that A > B !0 implies A' > B" > 0 for all

0 < a < 1 But this isnotnecessarilytruefora > 1

Thefollowing decompositionlemma is often useful and is anothertion of thestructure ofpositiveelements

applica-Lemma 2.2.14 Let W be a C*-algebra with identity Every elementA-eW hasadecomposition of theform

A = ajUj + a2U2 + a3U3 + a4U4

where the Ujareunitaryelementsof% and the aieC satisfy IaiI :!! 11A11 /2.PROOF It suffices to consider the case 11AII = 1 But then A = A, + iA2 with

A, = (A + A*)/2 and A2 =(A - A*)12i selfadjoint, IIA111 < 1, IIA211 < 1 A

general selfadjointelementBwith 11B11 < I can, however,be decomposedinto two

unitaryelementsB =(U+ + U_)/2 bytheexplicitconstruction U

+

= B +i'll- B2

As afinal application of theproperties ofpositive elements we consideranother type of decomposition. First let us extend our definition of themodulus If W isaC*-algebrathen A*A ispositivefor allA e%byTheorem2.2.11 The modulus ofA e% is thendefined by IAI = 1A*A IfA is self-adjointthis coincides with thepreviousdefinition Nownotethat if %contains

anidentityandAis invertible then A*A is invertible and its inverse ispositive.

It follows that IAIis invertible and IAI- "

= ,I(A; A Butonethen has

A = UJAJ.

where U=AIAI Moreover, U* U=I and U is invertible(U AIATherefore U is aunitaryelement of W and in fact lies in the C*-subalgebra

generated byA andA* ThisdecompositionofAis aspecial caseof the

so-called polar decomposition. The general polar decomposition concerns

operators on a Hilbert space and represents each closed, densely definedoperator A as a product A = V(A*A)1/2 of a partial isometry V and a

>0for all c.5.InHilbert spacetheorythislastpropertyisusuallytakenasthedefinition ofpositivitybutitisequivalenttotheabstract definitionbythefollowing reasoning.If the values of( , A ) arepositivethentheyare, in particular,real and(0, AO) = (A0, 0).Therefore the polarizationidentity

3

4k=O

demonstrates that (0, Aq) = (A0, 9) for all 0, oc-.5, i.e., A is selfadjoint But if

. Nowdefineanoperator V

onallvectorsof the formIA10 bythe action

VIA10 = A0

This isaconsistentdefinition ofalinearoperatorbecauseIA10 =0 isequivalentto

0 111A10 11 = 11A0 11andhenceA0= 0.Moreover, V isisometric because11VIA10 11

= A0 11 = 111A10 11.We may extend V to apartialisometryon.5bysettingitequal

to zero onthe orthogonal complement of the set I IA10; 0c-.51 and extending bylinearity.Thisyieldsthepolar decompositionofA,i.e.,A = VIA1.Thisdecomposition

is uniqueinthesensethat ifA = UBwithB > 0 and U apartialisometrysuch thatU9 = 0just for o orthogonal to the range ofB then U = V and B = I Al Thisfollows because A*A = BU*UB= B 2

and henceBisequalto the uniquepositive

square rootIAIofA*A.ButthenUIAI = VIAIand bothU and V areequalto zero

ontheorthogonal complementof therangeofIA1.Ingeneral,Vwillnotbeanelement

of theC*-algebra 91A generated byAandA*,althoughwehaveseenthat this is the

casewheneverAhasaboundedinverse.Nevertheless,in Section 2.4wewillseethat

Visanelement of the algebraobtained by addingto 91A allstrongorweak limitpointsofnetsof elements Of%A-

2.2.3 Approximate Identities and Quotient

Algebras

In Section 2.2.1 we gave examples of C*-algebras which failed to have an

identityelement and demonstrated that it isalways possibletoadjoinsuchan

element Nevertheless, situations often occur in which the absence of an

identityis fundamental and it is therefore usefultointroduce the notion ofan

approximate identity.

Definition 2.2.17 If 3 isarightideal ofaC*-algebraW thenanapproximate

identity of-1 isdefinedtobeanet' JE, I ofpositiveelementsE,,c-3 such that(1) JJE,,,JJ < 1,

(3) lim,,, I I E,,A - AI 1 0 for allAc- 3

'

A set I& is said to be directed when there exists an orderrelation,(X between certain pairs

of elements(x,#c- ?/ which is reflexive(a:!! a),transitive((X <#and 1,,implya < 1),

anti-symmetric (a < and#:!! aimply #=a)and when for eachpaira, c-& there exists a ysuch

that ot < 7and A net is afamilyofelements,of ageneralsetM, which is indexedbya

directed set J//.

The definition ofanapproximate identity ofa left ideal is similar but

con-dition (3)isreplaced by

(3) lim,, 11 AE,, - All = 0for allAc--3

Itisnecessaryto prove the existence ofapproximate identities

Proposition2.2-18 Let 3 bearightidealofa C*-algebraW-3possessesan

As each Aic-3onehas E_ F,,c-3 Furthermore JJEJJ < 1 and

(E,,,Ai - Ai)(E,,Ai - Ai)* (E,, - I)AiAi*(E,,- 1)

Proposition2.2.19 Let 3bea closed two-sided idealofa C*algebra%.It

followsthat3 is selfadjointandthequotient algebra %/3 definedin Section2.1.1 isa C*-algebra.

PROOF Let f E,,Jbeanapproximateidentityof 3 IfA e3 then 11A*E,-A*

IIE,,A -All >0 ButA*E,,c-3 and hence A*c3 because 3 is closed Thisprovesthat 3 isselfadjoint

To complete both the discussion of the quotient algebragiven in Section 2 1.1and the proofof the proposition we must show that the norm on the quotientalgebra,

IJAII = inf{IIA + III;Ic31,has the C*-normproperty. Toprovethiswefirst establish that

IJA11 limsup11A - E,,A11

lim inf11A - EAll

> inf{IIA + Ill;Ic-31 = 11,411.

The C*-normpropertyisthenaconsequence of thefollowingcalculation:

11,4112= limlIA - E,,A112

= limll(A - E,,A)(A - E,,A)*Il

= limll(I - E,)(AA* + I)(T - Ea)II

11AA* + Ill,whereIisanarbitraryelement of 3 Thus

11,jI12 < lI'j'j*II < 11,11111,4*11,

whichimpliesfirstlythat 11,411 = 11A*11 and,secondly,that

2.3 1 Representations

In the previous sections we partially described the abstract theory of algebras and illustrated the general theory by examples of C*-algebras ofoperators actingon a Hilbert space Next wediscuss representation theoryand develop the connection between the abstract description and theoperatorexamples. The twokey conceptsin this development arethe con-

C*-ceptsofrepresentationandstate.Thestatesof 91are aclass of linear functionalswhich take positive values on the positive elements of % and they are offundamentalimportancefor theconstructionofrepresentations.Weprecedethe discussion of thesestatesbygivingtheprecisedefinition ofarepresenta-tion andby developingsomegeneral properties ofrepresentations.

First let us define a *-morphism between two *-algebras W and 0 as a

mapping7r; Ac-WF +7r(A)c-0,definedfor allA c-% and such that

Noweach*-morphism7rbetweenC*-algebras%and 0 ispositivebecause

ifA > 0then A=B*B for someBc-WbyTheorem 2.2.12 and hence

n(A)= n(B*B)=7r(B)*n(B) >0

It is less evident that 7r is automaticallycontinuous

Proposition 2.3.1 Let % be a Banach *-algebra with identity, 0 a

C*-algebra, and 7r a *-morphism of% into0 Then n is continuousand

117T(A)II < 11AII

for all Ac-% Moreover, if% is a C*-algebra then the range 0,, = 17T(A);

Ac-% of7z isa C*-subalgebra of0

42

PROOF Firstassume A =A* Then since 93 isa C*-algebraand7r(A)c-0,one

has

11 7r(A) 11 =sup{IA1;Ac-u(7r(A))l

byTheorem2.2.5(a).Next defineP =n(l,)where1,denotes theidentityof W Itfollows from the definition of7rthatPisaprojectionin 0 Hencereplacing0bytheC*-algebraPOP theprojectionPbecomes theidentityI

,of thenewalgebra0.Moreover, n(91)s:-F8 Now it follows from the definitions ofamorphismand of thespectrum thatcz(7r(A))=uw(A).Therefore

11 n(A) 11 sup IA1;Ac-u,,(A)l A11

by Proposition2.2.2.Finally,ifAisnotselfadjointone cancombine thisinequalitywith the C*-normpropertyand theproductinequalitytodeduce that

11 7r(A) 112

=117r(A*A)II < 11A*A11 :!! 11A112.

Thus 11 7r(A) 11 < 11A11 for allA eW and7ris continuous

Therange 0,.,is a *-subalgebra of 0 bydefinition and to deduce that it is a

C*-subalgebrawe mustprove that it isclosed,under theassumptionthat W isaalgebra

C*-Nowintroduce the kernel ker7Eof7rby

of 91,, are the classes 4 ={A+I; I cker 7rJ and the morphism 7r induces a

morphismft from%,,onto!5,,bythe definitionA(,4)=7r(A).The kernel ofAiszero

byconstruction and henceir'isanisomorphismbetween%,,andF8,Thuswe can

define a morphism A` from the *-algebra 0,, onto the C*-algebra ' t7, by

A

Asuccessivelyoneobtains

11,411 = Ili-ImAvi < 11A(A)11 :! 11,411.

Thus 11A11 = 11A(A)11 = 11 7r(A) 11.Consequently,if7r(A,,)convergesuniformlyin 0to

an elementA,, then A.converges in 91,, to an element A and A,,=A(A)=7r(A)whereAis any element of theequivalenceclassA ThusA,,c07,andF8.,,is closed.Nextwedefine theconceptof*-isomorphismbetween C*-algebras.

A *-morphismnof %to 0 isa*-isomorphismif it isone-to-oneand onto,

i.e., if the range of7r is equal to 0 and each element of 93 is the image ofa

uniqueelement of W Thus a*-morphism7rof the C*-algebra% onto a algebra0 is a*-isomorphism if,and only if,ker-g =0

C*-Nowwe canintroduce the basic definition ofrepresentation theory.Definition2.3.2 Arepresentation ofa C*-algebra91 isdefinedto beapair (.5, 7r),whereSn is acomplex Hilbert space and7ris a*-morphismof % intoY(.5). The representation (.5, 7r) is said to befaithful if, and only if, 7E is a

There is a variety of rather obvious terminology associated with thisdefinition.The space.5iscalled therepresentationspace, theoperators7T(A)

arecalled therepresentativesof Wand, by implicitidentification of7rand thesetofrepresentatives,onealso says that7r is arepresentationof %onSv

The discussionprecedingDefinition 2.3.2 established that eachtion (.5, 7r)ofaC*-algebra% definesafaithfulrepresentationof thequotient algebra %,, = %/ker7r. In particular, every representation of a simple C*-algebrais faithful.Naturally,themostimportant representationsarethefaithfulonesand it is usefultohave criteria forfaithfulness

representa-Proposition2.3.3 Let (.5, 7r) be a representation ofthe C*-algebra %.Therepresentation isfaithful if,andonlY if,itsatisfieseach of thefollowing equivalentconditions:

(1) ker7r = {01;

(2) 117r(A) 11 = 11A11 forallAc-W;

(3) 7r(A) > Oforall A > 0

PROOF Theequivalenceof condition(1)andfaithfulness isbydefinition Wenow

prove(1)=>(2)=>(3)=>(1)

(1)=>(2) As ker7r = 01 we can define a morphism7r-' from the range of7r

into % by 7r-'(n(A))= A and then applying Proposition 2.3.1 to 9' and 7r

successively onehas

JJAJJ = JJ7r-1(7r(A))JJ :!! 117r(A)II :! JJAJJ

(2)=:>(3) If A > 0 then 11A11 > 0 and hence 11 7r(A) 11 > 0, or 7r(A):A 0 Butz(A)>0by Proposition2.3.t and thereforeir(A)> 0.

(3)=>(1) If condition(1)isfalsethenthere isa B c-ker7rwithB :A 0 and7U(B*B)

0 But JJB*BJJ ! 0 andas JJB*BJJ = JJBJJ'onehas B*B> 0 Thus condition(3)isfalse

A*-automorphiSMTofaC*-algebra91 is definedtobea*-isomorphismofintoitself, i.e., -c is a *-morphismof % with range equal to % and kernelequalto zero

:

TheforegoingargumentutilizingtheinvertibilityOfTimpliesthefollowing:

Corollary2.3.4 Each *-automorphism -r of a C*-algebra % is norm

preserving,i.e., 11 -r(A) 11 = 11 All forallAc-W

Nowweturnourattentiontovarious kinds ofrepresentationand methods

ofcomposingordecomposing representations.

Firstweintroduce the notion ofa subrepresentation.If(.5, 7r)is a

repre-sentationof theC*-algebra91 and.51isasubspaceof5then.51 is saidtobe

invariant, or stable, under 7rif7r(A)- ', S31 for all Ac U If51 is aclosedsubspace of .5 and P_5, the orthogonal projector with range .51 then theinvariance of.51 under7rimpliesthat

Note that theforegoingmethodofpassingtoasubrepresentation gives a

decompositionof7rin thefollowingsense.If.5, is invariant under7rthen itsorthogonal complement 51' is also invariant Setting -52 = -51j- one can

define a second subrepresentation (55,2,7r2) by 7r2(A) = P

.5 has a direct sum decomposition, S;,) = -51 (D 52, and each operator

7r(A) then decomposes as a direct sum n(A) = 7r,(A)ED n2(A). Thus we

write7r =

7rl ED 7r2and(15, 7r) = 051, 7rO ED0529

n2)-A particularly trivial type ofrepresentation of a C*-algebra is given by

n = 0, i.e., n(A)= 0 for all AeW Arepresentation mightbe nontrivialbutnevertheless haveatrivialpart.Thusif.50isdefinedby

.50 = f ; V/c-.5, 7r(A) =0 for all A c-WJthen fV,0 is invariant under 7r and the corresponding subrepresentation

nondegenerateif.50 = {01 Alternatively,onesays thatasetT1 ofboundedoperatorsactsnondegeneratelyon.5if

Definition 2.3.5 Acyclic representation ofaC*-algebra%isdefinedtobea

triple (.5,n,92), where (15, n) isa representation of % and 0 isa vector in.5which iscyclicforn,in.5.

In thesequel,if there isnopossible ambiguitywewill oftenabbreviatetheterminologyand say that Q isacyclicvector, orQ is cyclicfor 7r.There isa moregeneral conceptthanacyclicvectorwhich is also often useful If R isa

closedsubspaceofSv then R is calledacyclic subspacefor.5whenever theset

5,11

a C- I

of the representation spaces .5,, is defined in the usual manner' and one

defines the directsumrepresentatives

7E = ( 7r,,,

a e I

by setting 7r(A) equal to the operator 7r,,(A) on the component subspaceThisdefinition yields bounded operators 7r(A)on .5because 11 7r,,(A) 11 :!!

IJAII, for all occ-I, by Proposition 2.3.1 It is easily checked that (25, 7r)

is a representation and it is called the direct sum of the representations

(.5a, na),,, One has thefollowing result

Proposition 2.3-6 Let (.5, 7r) be a nondegenerate representation oftheC*-algebra 91 Itfollows that 7r isthe direct sum ofafamily of cyclicsub-

representations.

PROOF Let{Q,,I,,c-jdenoteamaximalfamilyofnonzero vectors in.5suchthat

(7r(A)Q,,,7r(B)Qfl)= 0for allA,B c-it,whenevera:A fl.The existence of suchafamilycanbe deduced withthe aid of Zorn's lemma Next define!5,,asthe Hilbertsubspaceformed by closingthe linearsubspace J7r(A)K2_A c-911.Thisis aninvariantsubspaceso we canintro-duce 7r,, by 7r,,(A)=

Pb,.7r(A)P5. and ilt follows that each 7r_,,Q is a cyclic

The finite subsets F of the index set I form a directed set when orderedbyinclusion and consists of those families0 (p = J(p,,Jof vectors such that 9_ ,C-.5,,and

representationofW But themaximalityof the JQ,,I,,c-jand thenondegeneracyof7r

implythat there isno nonzeroQ whichisorthogonaltoeachsubspaceS5,,and hence

(1) 5", 7r =(1) 7r_,

a I a I

The foregoing proposition essentially reduces the discussion of general representations to that of cyclic representations. This is of importancebecause there is a canonical manner ofconstructing cyclic representationswhich we will discuss in detail in Section 2.3.3 Thetype ofdecomposition

usedto reduce thegeneralsituationtothecyclicsituationdependsupon theexistence of nontrivial invariant subspaces. No further reduction ispossible

in theabseaceof suchsubspaces and this motivates thenext definition.Definition2.3.7 A set 1JW of boundedoperatorson the Hilbert space b isdefinedtobe irreducible if theonlyclosedsubspacesofS.5whichareinvariantunder the action of 9JI arethe trivial subspaces f0J and-5. Arepresentation (.5, 7r)ofaC*-algebraWisdefinedtobe irreducible if theset7r(W)is irreducible

on b

Thetermtopologicallyirreducible is sometimes used inplaceof irreducible.The termirreducible is defined bythe demand that theonly invariant sub-spaces, closed or not, are fOl and .5 Actually, the two notions coincide forrepresentationsofaC*-algebrabutwewillnotprove thisequivalence.Thereare twostandard criteria for irreducibility.

Proposition2.3.8 Let 9JIbeaseffiadjoint setofboundedoperators on theHilbert space.5 Thefollowing conditionsareequivalent:

(1) 9Nisirreducible;

(2) thecommutant9M'offl,i.e.,thesetofallboundedoperatorson.5which

commutewith eachA c-M,consistsofmultiples oftheidentityoperator;(3) everynonzero vector c-.5iscyclicfor9JIin .5,or931 = 0 and5 = C.PROOF (1)=>(3) Assume there isa nonzero0suchthat lin.span JA ;Ac-9311 is

notdense in -5.The orthogonal complement of this set then contains at least one

nonzero vector and is invariant under 9R (unless V= (0) and C), and thiscontradicts condition(1)

(3)=>(2) IfT c-IJ91'then T*c9JI'and, furthermore,T + T*C-101'and(T - T*)l

i c-9JI' Thus if IN' -A CT then thereis aselfadjointoperator SC-M'such that S5' - AT forany Ac-C.Asall bounded functions of Smustalso be in thecommutant onededucesthat thespectralprojectorsof S alsocommutewith 9N But ifEis any suchprojector

and 0a vector intherangeofEthen0 =E0cannotbecyclicand condition(3)isfalse

(2)=:>(1) If condition (1) is false then there exists a closed subspace A ofwhich is invariant under 9N But thenPqc-9J1' and condition(2)is false

We conclude this survey of the basic properties of representations by remarkingthat ifone has arepresentation (.5, n) ofa C*-algebra then it iseasy to construct other representations. For example if U is a unitary

C*-Algebras Algebras

operatoron .5and weintroduce7ru by 7ru(A) = U7r(A)U* then(.5, 7ru) isa

second representation. This type of distinction is, however, not important

so wedefine two representations (.51, 7r,) and 052, 7r2)to beequivalent, or

unitarily equivalent, if there exists a unitary operator U from .5, to -52

suchthat

7r,(A) = Un2(A)U*

forall A c-W.Equivalenceof 7r, and 7r2 is denotedby7r, _- 7r2

2.3.2 States

Although we have derived various properties of representations of a

C*-algebra W we have not, as yet, demonstrated their existence Thepositivelinearforms, orfunctionals,overWplayanimportant role both inthis existenceproofand in the construction ofparticular representations.Wenext investigate the properties of such forms We denote the dual of 91 by

91*, i.e., W* is the space ofcontinuous, linear functionals over 91, and we

definethenorm of anyfunctionalfoverWby

11 f 11 = supf I f(A) 1; 11A11 = 11.

Thefunctionals ofparticular interestaredefinedasfollows:

Definition2.3.9 Alinear functionalw overthe*-algebraW is definedtobe

positiveif

w(A*A) > 0for all A c-W A positive linear functional w over a C*-algebra W with11coll 1is calleda state.

Noticethatwehavenotdemandedthatthepositiveforms be continuous.ForaC*-algebra continuityis in factaconsequence ofpositivity,as wewill

C*-algebrais of the form A*A and hencepositivityofwisequivalentto(obeing

Theorigin and relevance ofthe notion ofstateis best illustrated byfirst

assumingthatonehasarepresentation (.5, 7r)of theC*-algebra W Now let

KIc-.5be anynonzerovectoranddefinewoby

wn(A) = (0, 7r(A)fl)for all A c-W It follows that con is a linear function over W but it is also

(%(A*A) = 117r(A)K2 112 >0

Itcanbechecked, e.g., fromProposition 2.3.11 and Corollary2.3.13 below,

that 11conil = 1whenever 11KIII = 1and 7risnondegenerate.Thus in thiscase"

wn isastate States of thistypeareusuallycalled vector statesof the sentation(.5, n) Althoughthisexampleofa stateappears veryspecialwewilleventually see that it describes the general situation Every state over a

repre-C*-algebra is a vector state in a suitable representation. As a preliminary

tofurther examination of the connection betweenstatesandrepresentations

wederivesomegeneral propertiesofstates

The basic tool for exploitation of the positivity of states is the general Cauchy-Schwarz inequality.

Lemma2.3.10 (Cauchy-Schwarz inequality). Let w bea positive linearfunctionaloverthe *-algebra91 Itfollowsthat

(a) (o(A*B) = o)(B*A),

'(b) I o_)(A*

B) 12 _< (o(A*A)o_)(B*B)jbrallpairsA,Be91

PROOF ForA,Bc-W and Ac-Cpositivityofwimpliesthat

(o((AA + B)*(AA +B))> 0

Bylinearitythis becomes

IA12w(A*A)+ a)(A*B)+Aa)(B*A)+ w(B*B) >0

Thenecessary,andsufficient,conditions for thepositivityof thisquadraticform in A

areexactlythetwoconditions of the lemma

As afirst application of this result wederive the following ships betweenpositivity, continuity,and normalization forfunctionalsover

forsomeapproximate identity {E I of

Ifthese conditionsarefulfilled, i.e.,ifwispositive, then

(a) (o(A*)= w(A),

(b) I w(A) 12 _< co(A*A)jj(ojj,

(c) jw(A*BA)j <(o(A*A)jJBjJ,

(d), 11(oll = sup{w(A*A), IJAII 11

forallA, Bc-%,and

I I(oI Ilim w(E.),

cz

where {E,,jisanyapproximate identity ofW

PROOF (1)=>(2) Let A, A21 be a sequence of positive elements withIIAJI 1. Now ifAi:-: L- 0 andJ]i i< + oo thenYi iAiconverges uniformly, and,monotonically,to somepositiveA andhence, by linearityandpositivity

Aia)(A):!i o-)(A)< + oo.

Since this istrue foranysuchsequence itheco(A) mustbeuniformlybounded.Thus

M

+

=sup co(A); A > 0, 11A11 < 1If < + oo.

But it followseasilyfromProposition2.2.11 that eachA c-91 hasadecomposition

3

A

n=O

withA,, !!0 and 11 A 11 :!! 1 Hence IIco 11 :! 4M

+ < + oo,i.e., cois continuous.Next letusapplytheCauchy-Schwarz inequality of Lemma 2.3.tOtoobtain

I o-)(AE,,)12 <o-)(A*A)co(E2):!! _ M+11A112co(E2).

Takingthe limitover cc onefinds

E., one also has 11coll :!! _lim,, co(E.)< 11coll Thus 11coll =lim,,, (o(E.) and the last

statementof thepropositionis established

(2)=>(1) We mayassume 11coll = 1 lf III has anidentityI then

III - E"211 < III - E,,II + III - EJI IIE,11andwehavelim,,,E'2 = 1 Henceco(I) = 1 If 'If doesnothaveanidentityweadjoin

oneand extend co to afunctional(Z) onit CI + 91by

C)(Al + A) A + w(A)

Because A - AE"2 = (A - AEJ + (A - AE,,)E,, we have lim,,A E"2 A. Usingthedefinition of the norm on A_,Proposition 2.1.5,we tnenhave

Co(Al + A) I A+ co(A) I = limIA(o(E"'2)+ (o(AE2)1

limsuplIAE '2 + AE '211 :!! IIAI +All

Thusin any case we may assumethat % hasanidentityand

ButA + iTT isnormal withspectrumin

a(A) + iT A A11 + iT

Hence

+ 2

11A + iTT 11 = p(A + iTT) 111 A j

SinceI co(A + V/1) + TIweobtain

But(o(T)= I andw(A*A)isreal and itisnecessarythat

(o(A*B*BA)< JIB112co(A *A)

Property (d)follows from(b)

11COl + (0211 = lim(o.),(E 2) +w,(E '2))

= lim.co,(E"2)+ liMOJAE"2)= 11(0111 +

11(0211-Finally,ifw, and(02are statesthenco = Aw, + (I - A)OJ2ispositivefor 0:!! A< 1

and 11co11 = A11co

111 + (1 - ) 11W211 = 1 Thuso)isa state.

Next remark that if W is a C*-algebra without identity element and

fl = C1 + % is the algebra obtained by adjoining an identity then every

o-) c-%* has an extension Coc-ft* defined by (7o(A + A) = Allo_)11 + w(A).This extension (Tj is usually called the canonical extension ofw and it is a

stateextension

Corollary2.3.13 Let % be a C*-algebra without identity andft the algebra obtained by adjoining an identity Further, let co be a positive

C*-functional over % and (b its canonical extension to ft. Itfollows thatCo is

positive and 11611 = 11o)JI. Moreover, if0)1, 0)2 are two positive forms and

(bl, C02 their canonicalextensionsthen

(0

1 + 0)

2 `: (01 + CO2

-PROOF Applying Proposition2.3.11(b)one estimatesthat

Co((AT + A)*(Al + A)) IA1211(J)11 + co(A) + Ao)(A*) + (,)(A*A)

0)111 + 11C0211 = 11COl + (0211,whichyieldsthelast statementof thecorollary

The property ofpositivity introduces a natural ordering of functionals

If (t), and 0)2 are positive linear functionals we write wl ! C02, or o),

-W2 > 0, whenever co,

-(02 is positive and we say that o_), majorizeS W2The properties of states with respect to this ordering will be ofgreat sig-nificance throughoutthesequel.

9 and 0 < A < 1 then (o = Acol + (1

a statewith thepropertythat w > Aco

1 and co > (I - 402Thus ifwisa convex combination oftwodistinct states then it majorizes multiplesof bothstates.It is natural to call a statepure wheneverit cannot

be writtenas a convexcombination of otherstatesand theforegoingremark

on majorizationmotivatesthefollowingdefinition:

Definition 2.3.14 A state co over a C*-algebra is defined to be pure if the

only positive linear functionals majorized by w are of the form Aco with

0 < A < 1 Thesetof allstatesisdenotedby E9,and the setof purestatesby

P14.

To conclude this sectionwederivesomeelementary propertiesof thesetsofstates EwandPA. As thesesets aresubsets of the dual W* of Wtheycan betopologized through restriction of any of the topologies of91* There are

twoobvious suchtopologies.Thenorm,oruniform, topologyisdetermined

by specifying theneighborhoodsofw tobe

T(W;,B) = {W';W,C-%*, 11W - W'11 < 81,where 8 > 0 In the weak* topology the neighborhoods ofw are indexed

byfinitesetsofelements, A,,A2,- ,A,,c-W,ande > 0 One has

0&((o;A15. .,A,; E) = f w'; w'c-W*, I w'(A j) - (o(A j) I < e, i = 1, 2_ ,nj

In practice it appears that the weak* topology is ofgreatest use although

wewill later haverecoursetothe uniformtopology.

Theorem 2.3.15 Let W be a C*-algebra and let B91 denote the positive

linearjunctionals over% withnormless thanorequalto one.Itfollowsthat

Bw is a convex, weakly* compact subset ofthe dual W* whose extremal

pointsare0 and thepurestatesP%.Moreover,B%isthe weak* closureofthe

convexenvelope ofitsextremalpoints.

ThesetofstatesEw isconvexbutit isweakly* compactif,andonly if,Wcontains an identity. In this latter case the extremal points of EW are thepurestatesP% andEwisthe weak* closureoftheconvexenvelope of P%.

PROOF B91 is a convex, weakly* closed subset of the unit ball %,* of%*, i.e.,

It follows thato) >_ A(o, and hence Aw, =

yaj for some 0 < y < I by purity But

1 = 110)11 = 4(0111 + (1 - *01W211 and one must have 11wj11 = I = 11(0211.Therefore

A= yand(o = w, Similarly,(.o= (02 and hence(,oisanextremalpointofB%

Supposenowthat(oisanextremalpointofBwandw=A 0 Onemusthave11o)11 = 1.Thus(oisa stateandwe mustdeduce that it is pure.Supposethecontrary;then there

is a state a), :A coanda A with 0 < A< 1 such that w > A(o,.Define(j,2 by 0)2=

((o - Aa),)/(l - A); then 11(1)211 =(11(oll - Ajja)jjj)/(1 -A)= I and (02 is also a

state.Butw = Awl+ (1

-A)(1)2andwisnotextremal,which isacontradiction.ThesetBw is the closedconvexhull of its extremal points bythe Krein-Milmantheorem This theorem asserts in particular the existence of such extremal points,which isnot atall evidentapriori

Finally,if % containsanidentityI thenEAisthe intersection ofB91with theplane (o(l)= 1 Thus the convexity, weak* compactness, and the generation pro-pertiesofE%follow from the similarpropertiesofB91.It remainsto provethatE%is

hyper-notweakly*compactif T0W and this will be deduced in Section 2.3.4

2.3.3 Construction of Representations

If (.5, n) is a nondegenerate representation of a C*-algebra 91 and Q is a

vectorin.5with JJQJJ = 1then wehave deduced in theprevioussectionthatthe linear functional

wn(A) = (0, 7r(A)Q)

is a state over % Thistype ofstate iscalled avectorstate.Nowwewantto

prove the converse. Every state is a vector state for some nondegenerate

representation.Thusstartingfroma state (o wemust constructation(.5., 7r.)of W anda vector0.c-.5.such that(o isidentifiedasthevector

representa-stateo-)Q_,i.e., such that

o)(A) = (Q., 7r,,,(A)Q.)for allA c-%

The idea behind this construction is very simple. First consider thedefinition of the representationspace .5 The algebra % isaBanach spaceand with the aid of thestate wit may be converted-into apre-Hilbert space

byintroduction of thepositivesemidefinite scalarproduct

<A, B> = o)(A*B).

Next define3 by

3. = A;A c-9A, o)(A*A) = 01,Theset3.is aleftideal of % becauseI c-3,,,andA c-%impliesthat

0 :! w((AI)*AI) !:_ 11A 11 2(0(1* 1) = 0

by Proposition 2.3.11, i.e.,AI c-3,,.

Now defineequivalence classes A, B by

OA = JA; A = A + I,I c-3,,,jand remark that theseequivalenceclasses also formacomplex vectorspacewhen equipped with the operations inherited from 91; OA + OB= OA+B,

*A = O.A Furthermore,this latter space is a strict pre-Hilbert space withrespectto the scalarproduct

( A, OB) = <A, B> = w(A*B).

It must, ofcourse, be checked that this is a coherent andcorrect definitionbut this is easily verified with the aid of Proposition 2.3.1t For example, (OA, OB) is independent of the particular class representative used in itsdefinition because

w((A + I,)*(B + 12)) = co(A*B) + w(B*Il) + w(A*12) + 0)(Il*I2)

= (o(A*B)

whenever11, 12 c-%. It iswell known thatastrictpre-Hilbert space may becompleted, i.e., linearlyembeddedas adensesubspaceofaHilbert space ina mannerwhich preserves the scalarproduct,and thecompletionof this space

is definedas therepresentationspace35,

Next letus consider the definition of therepresentatives 7r"'(A). First we

specify their action onthe densesubspaceofb,, formedby the vectors B,

Bc-91, bythe definition

nw(AVB z ABNote that this relation is again independent of the representative used forthe class Bbecause

7ccO(A)OB+I OAB+AI OAB 7rw(A)OBfor Ic-Zf, Moreover,each n,(A)isalinearoperator because

7r(,(A)(4B+ Oc) = 7r,,(A)OAB+C = OAAB+AC

= 4AB + OAC

= ),7r,,(A)OB +

7'co(A)OC-Finally, by Proposition2.3.11(c)onefinds

I I n,,(A)O B112 = (OAB, OAB)

It remainsto specifythevectorQ

If W contains theidentitywedefineQ by

QW = 01and thisgivesthecorrectidentificationofco:

is inthe closure of the set Let JEj beanapproximate identityofthen

by Proposition 2.3.1t and the desired resultis established

We havenow established theprincipal partof the followingtheorem.Theorem 2.3.16 Letco be a state over the C*-algebra 91 Itfollows thatthereexistsa cyclic representation (.5,n,Qj of91 such that

(o(A) = 7r,,,(A)Q,,) for all A c-% and, consequently, I I f2.112 =11coll = 1 Moreover, the repre-sentation isuniqueupto unitary equivalence.

PROOF Theonlystatementthat wehave not as yetproved istheuniqueness. Bythiswe meanthat if 7r,,,', is asecondcyclicrepresentation such that

w(A) = (Q.', 7r,,(A)Q,,')for allA c-% then thereexists aunitaryoperatorfrom.5.onto.5",'such that

U

-'7r.'(A)U = 7c,,,(A)for allA c-W,and

Un = Q.'

Thisis,however,establishedbydefiningU through

andnotingthat

U7r,,(B)Q.)

= (i)(A*B) =(7r,,(A)Q,,, n,,)(B)f2,j

Thus U preserves the scalar product and is consequently well defined It easilyfollows that theclosure of U isunitaryand has all the desiredalgebraicproperties

We omit the details

Corollary 2.3.17 Letcobe a state over theC*-algebra W and 'r a morphism of91 which leaves(o invariant, i.e.,

*-auto-(o(T(A)) = co(A) for all A c-% Itfollows that there exists a uniquely determined unitary

operator U,, on the space ofthe cyclic representation (Sv ,,,7r,", Qj

con-structedftom(o,such that

U 7r.(A) U,,- ,co 7r,,,(-r(A))