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1 72-10,168 VAN CASTEREN, Jhaes A., 1943 GENERALIZED GELFAND TRIPLES. Uiversity f Hawaii, Ph.D., 1971 Mathematics Uiversity Micrfilms, A XEROX Cmpay. A Arbr, Michiga THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED

2 GENERALIZED GELFAND TRIPLES A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF BAHAII IN PARTIAL FULFIL.11ENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS SEPTEMBER 1971 By Ja A. va Castere Dissertati Cmmittee: Rudlf A. Hirschfeld, Chairma h. S. :ar Lawrece J. '''e.lle Bharat Kiariwala

3 PLEASE NOTE: Sme Pages have idistict prit. Fi lmed as received. UNIVERSITY MICROFILMS

4 iii ABSTRACT I the dissertati. which maily deals with camutative semi-prime algebras ad represetatis theref. we first examied the class f the s-called regular ideals. Defiitic 1. ~ A ~ ~ camutative semi-prime e.lgebra. ~ ay subset S ~ A, ~ defie lli ideal SC &. SC = {a A. ax = 0 fr all xes}. ~ ideal I i!l A ~ called regular II I = ICc. I!!. lvig peratis: the class f the regular ideals, we itrduced the f1- where (Iv) is ay subset f B ad I ay elemet i B. We shwed that, uder these peratis, B is a Blea algebra; sice fr ay subset (Iv) f B the itersecti l v belgs t!!o, it is cmplete; sice H,(V Iv> = V(IAI), where (Iv> is ay subset f B ar.d I is ay elemet f ~, it is distributivee Defiiti 2. ~ a*ebra A satisfies the cutable chai cditi (c. c. c. ), if. ~! ~ assertis ie.. ~ fllwig therem hlds i A. Therem 1. 1!:!!:. fllwig assertis ~ eguivalet. 1. Every disjit fami1,y (b v ) (~. ~ :# " implies b\.lb V = 0)! -zer e lemets k cutable ; 2. Every itersecti lv E!. regular ideals Iv ~ cutably accessible (~. I.uc:rc: t:ju..::;i.:s a cul;.able subfamily (Iv ) such that iii" = iil v ). I the special case where A = C(X). the algebra f the ctiuus cmplex-valued fuctis e the ccmpact Hausdrff space X, we were iterested whether r t there exists a gt~ictly psitive ma"..., , ~. a prbability measure with the prperty t:hat every -vid pe

5 iv subset f X has psitive measure. I this cecti we gt the fllvig result. 'Iherem 2. ~ fllevig assertis ~ eguivalet. 1. There exists!:. strictly psitive prbability measure E.!!. X; 2.!!. every -vid regular pe ~ 00.!a. X there exists!:. buded regular psitive measure \.1 0 ~ 00' ~~ \.10(0) i:l psitive!e every rme set 0, which is dese i, mrever, X satisfies _the -~ c. c. c. (~. every disjit fami~ f -vid pe sets is cutable) I rder t cclude 1. frm 2. a slight~ weaker cditi is suffidet: X satisfies the c.c.c. ad fr every -vid regular pe set 00 tgether with ay subcllecti f {O, pe ad dese i 00} with the prperty that the iterir f ay cutable itersecti O' O i, agai belgs t.s there exists a measure \ ' such that \.10(0) is psitive fr every member f. I terms f Blea algebras we prved the fllwig. Therem 3. ~ B ~ ~ caple~ distributive Blea algebra. ~~ lvig assertis ~ equivalel 1. There exists ~ buded strictly psitive measure lj B (b.!.. p :f. 0, P E B, implies lj(p) > 0), 2. E2!. every elemet P'!!!' B there exists ~ buded psit!ve measure \.I P = VPi' mrever, B satisfies the c.c.c - 1_,.L',_ 0-.. \1-'i J ~ WUJ.l;lJ.

6 v I what fllws we assue that the represetati U: A - L(F). where F is ay lcally cvex vectr space. has the prperty that fr every ideal I i A the prjecti exists ad is ctiuus. Amg thers we prved the fllwig results. Therem 4. ~ A C(X). where X.!.!l ~ lcalj,y ccmpact Hausdrff space. which ~ ~ cutable ~.!2!:.ili. tplgy. ~ F ~ ~ rmed vectr space!!!!i ~lli represetati U: A - L(F) ~ faithful. ~ there exists!!!l. elemet ~ F' ~ ~ ~ ideal {xe:a. <U(x) f.~> = a fr all f i F} reduces l2. {a}. prvided ~.!2!:. every f e: F ~ ~ F' ~ mappig x'-' <u(x)f.~>. x e: A.f!!. ctiuus. Therem 5. Assume ~ ~ vectr space F ~ ~ writte ~ ~ tplgi- _ cal..=di... -.r... e-.ct... sum f cmplete metrizable vectr spaces H Let the spaces H 'V -- - V ~ miimal ~ 2 sese ~ there 2.!! i exist prper clsed ivariat subspaces H f H fr which the represetati U _is faithful. Assume that - -v ~ semi-prime algebra A satisfies ~ c.c.c ~.f2!:..~ \I there ~ ists a vectr f such that U(A) f is dese i H Assume. _i additi. -- v v - v ~ ~ spaces H\I ~ Baach spaces ~ ~ U(I)f = {a}, where f e: F ~ I ~ ideal i A.!2!:. which ICC = A. imply f = a. ~!2!:.~ \I there ex- I... \.L ; i... \.. U"I, ::;Ucll thai; li(a):<jl is w--dese i v v- - 1l#\I

7 vi TABLE OF CONTENTS ABSTRACT iii CHAPTER I. L1iTRODUCTI ON CHAPTER II. BOOLEAN ALGEBRAS AND IDEALS 1. A Blea algebra f a certai class f ideals i a rig Regular ideals The cutable chai cditi 15 CHAPTER III. SCtfE CCt.fMENTS ON STRICTLY POSITIVE FUNCTIONALS 1. Prelimiary remarks Regular 1\mctials ad rmed algebras Strictly psitive fuctials Blea algebras ad strictly psitive measures 40 CHAPI'ER IV. GENERALIZED GELFAND TRIPLES 1. Represetatis f semi-prime algebras 2. The geeral situati 3. The situati where U(I)F is dese i U(ICC)F.. BIBLIOGRAPHY

8 1 CHAPTER I IN'ffiODUCTION I the preset wrk we shall geeralize the s-called Gelfad triples fr Hilbert spaces t arbitrary lcal~ cvex vectr spaces. I terms f a give peratr (r a fami~ f peratrs) defied a Hilbert space H, e fte arrages fr a triple H O ~ H '- H~, where H is a certai lcally cvex vectr space (which m83' be a uclear space, a Baach space, etc.), which is dese i H ad fr which certai ivariace cditis hld. The space H~ is the tplgical dual f H Fr ccrete examples see ~. Ju. Berezaskii [2], I. Gelfad ad thers i [10] ad [11], R.A. Hirschfeld [14], K. Mauri [18]. We shall csider a represetati U: A -- L(F), where A is a cmmutative semi-prime algebra (Le. a 2 = 0 implies a = 0) ver the cmplexes, F ay lcally cvex vectr space ad L(F) the algebra f all ctiuus edarphisms f F. I A we csider a certai Blea algebra f ideals. May f the prperties preset i case A is geerated by a Blea algebra f idemptets remai valid r ca be frmulated i tes f this Blea algebra f ideals. Our ultimate aim is t arrage fr F' C-..+- F' ad F <-+ F' where F0 is U-ivariat (i:.!.. U(x)F0 C F0 fr all x c A) ' ad the imbeddig J: F 0 -~ F' has the prperty that U(x) I Jf = JU(x) f fr all f c Fa I Chapter II we shall ivestigate the prperties f semi-prime rigs.

9 Mre specifically we are iterested i the class f the s-called regular ideals; see Defiiti I Chapter III we clseq examie the 2 "simple" situati, where A l:i F I: C(X). the algebra f a.ll ctiuus fuctis a capact space X, ad U: A -+ L(F) is defied by U(f)g = fg fr all f. g A. I rder t btai the ijecti F -+ F' we eed the,ti f a strictl,y psitive measure; see Therem Fially, i Cha).: ter IV we shall csider the gee:l:'al 8ituati. The measure-theretical tls we eed are take fr [9] ad [ 13]. Fr the thery f lcally evex veetr spaces we use [16] ad [23], where a great may results e (partiall,y) rdered vectr spaces ca be fud t. We emply the stadard prperties f Baach algebras as set frth i [21],'Uld [22]. A treatmet f lcally cvex algebras ca be fud i [20] ad [26]. Fr a survey f the prperties f Blea algebras see (24J. Fr prperties f (geeralized) spectral ad/r scalar peratrs we meti [4]. [5] ad [171 ad the refereces give there.

10 3 CHAPTER II BOOLEAN ALGEBRAS AND IDEALS 1. A Blea algebra f a certai class f ideals i a rig. Thrughut the sequel A stads fr a camutative rig (with r withut idetity). The preset secti is devted t the cstructi f a "caical" Blea algebra! f ideals I i A. N tplgy A wi 11 be eeded fr the time beig. Give ay set SeA, we will write SC = {ae:a, as = {a}} fr the aihilatr f S i A. (The superscript c is remiiscet f settheretical caplemetati.) It is clear that SC will be a ideal i A (psl:,ibly imprper) ad that S C SCC. We w impse the fllwig stadig hypthesis A: Fr every ideal I i A we have I~Ic = {OJ. Recall that a cmmutative rig is semi-prime if it has ilptets 1. Prpsiti ~ fllwig prperties ~ equivalet: (i) (ii).e2!:. every ideal I 1!!. A ~ ~ laic = {a}; ~ every ideal I i A, 12 = {a} implies I = {a}; (iii) ~ eveey elemet b ~ A, b 2 = a implies b = O. Prf. (i) => (ii) => (ii). If 1 2 = {a}, the I C Il c = {a}. (i). Fr ay ideal I i A, we have (II C }2 = {a}, s Il c = {a}. (i) => (iii). Suppse b 2 = O. Csider I = ba. The ba belgs t laic = {a} fr all a A ad s ba = {a}. Hece b AAA c = fol. (iii) => (i). If b C I~Ic, the b 2 = 0, go b = O. Fr mre ifrmati semi-prime rigs, see ~. [191. We w adpt the fllwig defiitis Deiti A ideal I i A fr whic.~ I = ICC is called regular.

11 Defiiti ~ 2ll2!:.. ~ family f ~ ideals I (prper.2!.~) i A, which ~ regular. We shw that all aihilatrs belg t ~. 4 Prpsiti !:2!:. ay subset SeA ~ ~ SC e: B; {a} ~ A belg ~~. Prf. 'le must prve that SC = SCCC. The bvius iclusi S c. SCC, implies SC c sccc. Cversely fr ay b e: SCCC we have bs cc = {a},whece bs = {a}, s that b e: SC. The remaiig statemets are bvius. Befre we itrduce peratis i ~ we agree up sme tati. If {Iv' v r} is a family f ideals i A, the EIv stads fr the ideal f all fiite ccmbiatis E~=l ai' where ~ e: UIv fr i = l,,. If I ad I are ideals i A, II:S is the ideal f all fiite cmbiatis 1 2 f the frm Ei=l ~b i, where a i e: II ' bi e: 1 2 fr i = 1,,. A similar tati is used fr the "prduct" III2".I m f ideals II,,~. I B we itrduce tw peratis: Fr a arbitrary family (Iv), where v waders ver sme idex set (which will t be metied) ad where all ~ belg t ~~ we defie III.,; = Iv, VIv =(~)C = (EIv)CC. It will be see that fr these peratis ~ becmes a cmplete Blea algebra. Mrever we will tice a strikig similarity with rdiary set thery. I rder t shw these prperties we will ee the fllwig lemma. Lea ~ II,,I~.!. fiite umber 2! ideals i A. The (II :s...i) cc :.. (li l ) cc = -;CCrL.."~I~C ~ secd equality eed ~ hld f9.!: ifiitely ~ ideals.

12 Prf. It will be sufficiet t prve the statemet fr = 2. Sice - always, 1 1 c 1,,1 c I~CrU~C, we have {I I )CC c {I f"li )CC c I 2 l 2 5 {I~c"I~C)CC. By the equality I~c"I~C = {I~ + I~)C ad by Prpsiti we get {ICC"Icc)cC = {I C + IC)cCC = {I c + IC)c = Icc"I cc S there remais t shw that It:(\I~C c{i I I 2 )CC, r equivaletly {I I I 2 )C C(I~C"I~C)C. Let a {I 1 I 2 )C. The all C I~ ad s {all )f\i~c = {OJ. Fr b ay elemet f III~C we have ab is a elemet f {all )f"li~c = {Ole S a (III~C)C, cc c (CC) cc {l cc ICC whece ai 2 c II. Thus ai 2 Il = O. Next let b belg t II 11 2 The ab E {a.i~c)iii~c = {Ol. Frm this we fially ifer a {I~cf\I~C)C. Next we will give a example i which we will see that the asserti (OI)CC = llig c, des t hld fr cutably may ideals I i A. I,at A = C[ 0,1], the rig f all ctiuus cmplex valued fuctis [0,1]. T each ratial umber r, 0 < r < I, we assig the ideal I r = {fea, f{r) =OJ. The l r = {fea, f{ r) = a fr all ratial umbers r} = {a }. S But (Or )CC = {a}cc = {O }. r I C = {gea, gf = 0 fr all f Irl = {a}. r Hece, 1::::::- At frm which we see Ice = A. r - r I the fllwig statemet we cllect sme f the prperties f B.

13 6 Therem ~ peratis A ~ V satisfy ~ fllwig rules: ( i ) (~ E!!. caplemets")!2!:. (I)!!!. arbitrary subset!~~~ c (a) AI VIc " " E: ~, (AI) =!, (b) VI" ~, (VI)C (ii) ~ "Distributive lavs") c = AI" B. :E2!:. (I)!!!. arbitrary subset.2!. ~ ~ I ~, ~ ~ (a) Iv(AI) = A( IvI), (b) IA(Vl) = V(lAI). ~ family ~ 1.:!. ~ Blea algebra,.!2!. ~ ~ (iii) Q (= {O}) ~ 1. (= A) belg l2.!. ( i v).e.2!. every e lemet I B there exists!:. uiquely determied elemet I ~, amely I = I C, satisfyig IAl = Q. ~ IVl = 1. Prf. (i)(a). We will prve that AI" = (EI~)c. I virtue f Prpsiti 2.1.4, this will shw AI v B. Fr a AI" = l v we have, sice I" = I~C, C c ai v = {J fr all ". S a(rl v ) = {J, whece a (EI~)c. Cversely, let a (El~)c. The a(ei~) = {J ad s bi~ = {J fr all ". Hece a I~c = Iv fr all v, whece a r v = Al v (i)(b). The first prperty fllws frm Prpsiti 2.1.4, the secd is a applicati f (i) (a). ~u -..-ii""t-uc uf' U.}( 8.J, (0 J it is sufficie~ ~ prve e f the equalities. Let us prve the secd e. Let I ad (I ) belg t B. v The it is easy t verify that

14 7 Up takig secd aihilatrs we get: A applicati f Lemma t the lef't-had side yieids : ICC{~Iv}CC = {~{IIv}c}c. Applyig the same lemma agai we see: {IIv}c = {IIv}cCC = {Icc~I~C}C = {I~Iv}C. Hece implyig (ii}{b). Asserti (iii) is bvius. {iv}. Give ay I ~. let 1 0! be such that IAI =2.. IVI = 1. Frm 1111 = I hi = a we ifer I 0 0 clude ICtU~ = {ICrU~}CC = A C = {O} ad thus e I C Fra {IcflIC}C = IVI = 1 we c 0 I C c I~C = 1 0, Hece 1 0 = I C Remark 1. I terms f lattice thery ~. tgether with the peratis A ad V. is called a Bruweria lattice. See [3] fr this ad related tpics. Remark 2. We did t use the fact that elemets f A have egatives. Fr example. we may apply the results f this secti t a ce A f psi tive fuctis. 2. Regular ideals. I the preset secti we first shall assciate the regular ideals (I = ICC) f a algebra A f fuctis a pit set X. t a certai Blea algebra f subsets f X. We ext address urselves t the mai tpic f the preset wrk. ~. the regular ideals belgig t a Blea algebra f idemptets. The results are useful i the spectral thcc:'".r ~ls.ted t a Dl -ail alg~br&. r prjectis defied a vectr space.

15 8 Our startig pit will be the bservati that ay subset f a cap1etely regular space is pe if ad ly if fr every x :O there exists a buded ctiuus real-valued fucti f X such that (i) f( x) # 0, (ii) f = utside f 0. Nw let X be sme pit set, K a field, ad A a rig f K-va1ued fuctis X. Usig this idea, we will defie pe sets i X. A subset f X is said t be hk-pe if fr every x 0, there exists a fucti f A such that (i) f(x) # 0, (ii) f = a utside f 0. It is easy t veri fy that a arbitrary ui f hk-pe sets is agai hk-pe. Mrever the itersecti f fiitely may hk-pe sets is hkpe. Let a,,0 be hk-pe sets ad x O. Sice fr each i, O. is 1 ~ ~ hk-pe, there exists a fucti f, such that f.(x) # 0 ad f. vaishes... ~ ~ ff 0i. Let f = fl ' f The f( x) 'f 0 ad f = ff i It is als easy t verify that fr each f A, the set {x X, f(x) # O} is hk-pe. Mrever if L is a arbitrary subset f X the the hk-c10sure f L csists f all pits x X, fr which f( x) = 0 fr all f which vaish L. I a frmula: L = {x X~ f(x) = 0 fr all f A fr which fl L = a}. The hk-tplgy is remiiscet f the classical hull-kerel tplgy the maximal ideal space f a cmmutative Baach algebra; see [15]. I fact. if the abave fu ct.;,..; "!e; _t._ 0!2 x: b e.ppe!'2s '"Q.."_,, l-. _, '1-..!.A... ~'- t.j.4 U with X as its maximal ideal space, the the hk-tp10gy itrduced abve is readily verified t cicide with the hull-kerel tplgy (whece the tati hk-tplgy).

16 9 Fllwig stadard termilgy' (cf. P.R. HaJms [12]) a subset <: X will be called a regular pe set fr the hk-tplgy if = iterir O. As shw g. the family f regular hk-pe sets is a Blea algebra fr the peratis 01 A0 2 = 1 1"' V0 2 = (01v02) I I where 0' is the caplemet f i X. I the ext sequece f lemmas ad therems we will establish a e-t-e crrespdece betwee regular ideals i A ad regular hk-pe sets i X. Lemma ~ P ~!!; subset EL A ~ ~ U = u.!.:!. hk-pe. U {X X. g(x) ~ g P C a}. Prf. (ii) flu = <=> f pc. (iii) U = U {X X. g(x) ~ l. g PC (i) The set {x X. g(x) ~ } is hk-pe fr g A ad the arbitrary.mi f such sets is pe. (ii) =>: Let flu = ad g P. Fr x U. we have f(x) = ad s f(x)g(x) = 0. Fr x t U. ~. g(x) = a fr every g P ad agai we have f(x)g(x) = 0. Hece f pc. <=: Let f pc ad x U. ~. g(x) 1 a fr same g P. The. sice f pc. f(x)g(x) =. S. sice g(x) ~ O. f(x) = 0. (iii) By the abve remarks U = {XEX. f(x) = a}. where the itersecti f is take ver all f fr which flu =. By (ii). U= {XEX. f(x) = a}. fep C Thus U' = X\u = U hex. f'(y) '! L fep C ~ ~ P ~~ subset f A. The P = pcc <=> P = {fea. fl = a}. where ~ ~ hk-pe subset f X. Prf. =>: Csider a = U gep C f X ad fl = 0 if ad ly if f E pcc = P. {XEX. g(x) 1 a}. The is a hk-pe subset

17 <-. Let P = {fe:a, ri 0 = O}, where 0 is a hk-pe subset f X. Fr every 10 x e: 0 there exists a fucti f e: A such that f(x) :I- 0 ad f = 0 ff O. S if I detes the ideal I = {fe:a, f = 0 ff l, 0 = U {xe:x, f(x) :I- l. fe:i Thus, by (ii) f Lemma 2.2.1, fl = 0 if ad ly if f e: IC. Hece P = IC. Thus, by Prpsiti 2.1.4, P = p Cc Lemma 2.2<3. ~ subset 0 2! X!!. ~ regular hk-pe ~II~~ g there exists ~regu1ar ideal I such that 0 = U he:x, f(x):1- le -- fe:i Prf. (sufficiecy) Let 0 = U {xe:x, f(x) :f. Ol' where I = ICC. fe:i The, by Lemma 2.2.1, 0" = U {xe:x, f(x) :I- } = O. fe:r cc (ecessity) Let I = {te:a, {xe:x, f(x) :I- O} co} where 0 is regular hkpe. Sice 0 is hk-pe, 0 = U {xe:x, f(x) :f. a}. By Lemma 2.2.1, fe:i I C = {fe:a, f = 0 O}. It fllow's that I C = {fea, f = 0 a} = = {fe:a, {xe:x, f(x) :I- l cx\o}. Sice X\O is hk-pe, we get X\O= U {xe:x, f(x):1- O}. Agai, by Lemma 2.2.1, f e: ICC if ad ly if fe:r c f vaishes X\O. Hece ICC = {fe:a, {xe:x, f( x) # 0 le}. Sice fr each f e: A, {xe:x, f( x) :I- a} is hk-pe, we ifer cc ~ I = {fe:a, {xe:x, f(x) :f. O} c::. It(OJ}. Sice 0 is regular hk-pe, 0 = It(O), whece I = ICC. Icidetally we als prved Lemma A subset P.! A ~ ~ regular ideal i A if ~~II there exists ~ regular hk-pe subset 0 2f. X ~ ~ As a csequece, we btai (tati frcm [12])

18 11 Therem ~ X ~ ~ pit ~, K!!. field ~ A!!. rig.2!. K- valued fuctis!!. X. ~ X ~ supplied ~ ~ hk-tplgy. ~ li ~ ~ Blea algebra 2!ill regular ideals!!!. A ~ 13 ~ ~ Blea ~ gebra..2!. ill regular hk-pe ~.!ll X. - ~, there exists.! mappig u: li ~ B ~!!. mappig v: B ~ li, ~ ~ yu = idetity m.li ~ uv = idetity.q!!. B. Mgrever if I, II ad 1 2 belg ~~, ~ u(i 1 Al z ) = u(i 1 )AU(I 2 ), u(i 1 VI 2 ) = u(i 1 )Vu(I 2 ) ~ u( I C ) = u(i)'. Similarly, v ~!!! prperties: v(oia02) = v(oi)av(02)' v(olv0 2 ) = v(oi)vv(0 2 } ed v(o') = v(o)c, where 0, ad 0 are regular hk-pe sets i B Prf. Defie u: ~ ~ B by u(i) = U {X X, f(x) ~ OJ, where I ~, ad f I defie v: B ~~by yeo) = {f A, {X X, f(x) # } co}, where 0 B. The, ideed, by Lemma 2.2.3, u maps ~ t Bad vu(i) = I fr all I B. By Lemma 2.2.4, v maps B t ~ ad uv(o) = 0 fr every 0 E: B. By virtue f these facts ad sice B ad B are Blea algebras, it will be sufficiet t prve that u(i 1 AI 2 ) = u(i1)au(i z ) fr all 1 1,1 2 B ad that u( I C ) = u(i)' fr all I E: lie By defiiti It is easy t veri fy that U {XE:X, f(x) # O} = fe:i{'1i z wlleee U {xe:x, f(x) ~ J. f I1()I Z U {XE:X, f(x) # O} U {XE:}', f(x) # J, fe:i} fe:lry This hlds fr all II ad 1 2 E: B.

19 12 If I ~. the by defiiti u(i) = f~i{j(f;x. f(x) '# O}. S. by Lemma u(i)' = ~IC{X X, f(x) :I- O} = u(i c ). Ntice that if A = C(X). the algebra f all ctiuus caplex valued fuctis the cmpact Hausdrff space X. the the hktplgy fr X cicides with the usual tplgy. Next we shall. i a atural way. cstruct a algebra f "simple fuctis" belgig t a Blea algebra. Let B be a Blea algebra uder the peratis A. V ad '. Its elemets will be deted by P. q Let K be a field with members A. ~ Let "5 be the set f all frmal fiite cabiatis f disjit elemets i B. Le. a elemet f e: 5 is f the frm f = L. A. p; ~=l ~... where A1 A e: K. Pl P B ad PiAPj = 0 wheever j f i. Frmally. we defie a scalar multiplicati. a multiplicati ad a additi as fllws: the Af l = Li~IAAiPi' fr all A e: K. m f l f 2 = L i =IL j =IAi1J j Pi!l.qj. ad f l + f 2 = Li~lrj~l(Ai + 1Jj)Pi Aqj + r. A.p.Aqt' Aq' ~=l :l. 1. m We call a elemet f = Li A.p. e: ~ trivial =1 ~ ~ J Pi = 0 ad if Pi :I- 0 implies Ai = 0 fr i = l. A elemet + L m 1J qj Ap' A lip I j=l j 1 if Ai f 0 implies f l = Li~IAiPi is said t be equivalet t f 2 = rj~l~jqj' tati f l ~ f 2 if the elemet Li~lrj~l(Ai - ~j)piaqj + Li~lAiPillqlA A~ + Lj~I(-1Jj)qj P~II IIP~ is trivial. This relati is a equivalece relati ideed.

20 13 If we d t distiguish betwee elemets i -5 ad their equivalece classes. the (scalar) multiplicati ad additi, defied abve, makes S 2 ~ I.v it a algebra ver K. The class f the trivial elemets will becme the zer elemet i S ad (-1) f will be the egative f f fr each f E: S. Up idetifyig p ad (the class f) l.p fr each elemet p E: B, B is i a atural wa:y a subset f S. Uder these idetificatis we have fr istace: pq ::: paq, p + q - pq = pvq. p + p' ::: e. where p, q E: B ad e is the idetity i B. Example 1. If B is a Blea algebra f prjectis defied a vectr space ver K. The S is the algebra f peratrs spaed by B. Example 2. If B is a Blea algebra f subsets f sae pit set X, the S is (isarphic t) the algebra f all simple K-valued fuctis spaed by the characteristic fuctis f members f B. Example 3. Let B be the Blea algebra f the regular pe sets f a tplgical Hausdrff space X. Let ~ be the cllecti f all K-valued fuctis f the frm r AiXO.' where ~r every i. 0i is a pe set i i=l ~ X, X its characteristic fucti ad where all Ai belg t K. 'I'w' fuc- i tis f 1 ad f 2 i S are said t be equivalet, deted by they cicide scme pe set. which is dese i X. The it is readily verified that S = -S/", is iscmrphic t the caical algebra. as cstructea abve. belgig t B. A Blea algebra B is said t be cmplete if fr every decreasig family (Pa) c B, its meet Ara. exists. It is called distributi'fe if fr every decree..sig f~ily (P a ) ad evc'i".i elemet pcb. pvj\pa

21 14 We will prve that. fr cmplete distributive Blea algebras. a ideal I C S is regular (i.e. I = ICc) if ad ly if I is f the frm I = pst where p belgs t the uderlyig Blea algebra. We first prve the fllwig lemma. Lemma ~ B ~!. cmplete distributive Blea algebra. S as abve. ~ ~ (Pa) ~!. decreasig family.!. elemets i B. The p S = (Ap )S. - a a = ~PePa = ~Pa = P S. if f S. the pf = PeP f fr all e. whece PS c p S. Cversely let g = E.lA.r. PaS. We will shw that pg = g. a ~= ~ ~ whece (p S) cpos. We may assue that a A. ". 0 fr all i. ~ '!he pg = Ei~lAiPOri = Li~lAi(Ap )r i = Ei~lAiA(Pari). Sice a g = Ei~lAiri PaS, we certaily have that g PaS. Hece. there exist cstats ljj. that L Lr. i=l ~ ~ A r ~ ~ Thus = O:i~lAi ri hi = m a r i = Lj=lPaqjri' j = l m tgether with elemets qj. j = l.m such m a a = Ej=llJjpaqj ad such that j ". k implies Multiplyig bth sides by r i a a ad by qj we have Ai riqj 'f a ~ h a, Hece. ~ Paqjri r O. we see t at 1J j -. I\i ad s m a a _ m a Ej=llJjPaqjri - Lj=lAiPaqjri whece P r. = r. a ~ ~ a a qjqk = O. a a = lj jpaqjri S we have that pg = Li~l AiA(Par i ) = Ei~lAiAri = Li~lAiri = g. Therem ~ B ~ S ~ ~ ie. Lemma A ideal I C S is regular g ~ ~ g I = ps!.!:. ~ P B. Prf. (sufficiecy) Let I = pst where p B. The f Ie if ad ly if fp = 0. r equivaletly. f = f(e - p) = (e - p)f. Thus (ps)c = (e - p)s. whece (ps)cc = ((e _ p)s)c

22 15 (ecessity) We will apply Zr's lemma. Csider a f~ily f icreasig idem.ptets (P a ) c I = ICc. Sice B is a caplete Blea algebra. we have ~(e - P(l) = e - P' fr sae P B. Thus by the previus lemma we have that ee - p )8 = (e - p )S. a a 0 We will prve that p I ad that p p = p fr all (I. a 0 a We have I :;) U{pS, P I, P B} ad slc c {( ps )c, P I, p B} z{(e - p}s, P I, p B} (as abve) c ( e - P }S = a fr which we see that P I C = that p(i(e - p) = 0 r PaP (e - p }S, {O} ad s P cc I = I. Mrever it fllws., Pa fr all (I. Csequetly we may apply Zr r S lemma, t the effect that there exists a maximal elemet p It\B. SuPpse there exists a elemet f I, f ips. The, by assumpti, the elemet f is f the frm f c ri~laipi' where Ai ~ 0 ad PiPj = 0 wheever j ~ i. Sice, fr every i, p.f = A.p. it fllws that every Pi I, 111 sice f t pa, at least e P j ips. Csider q = pjvp =P j + P - PjP. The q I ad q :f p ad pq = p. Hece P is t maximal, which is a COI'- tradicti. 3. The cutable chai cditi. I the fllwig chapters we will eed a certai cutability prperty f the rig A. We aim t geeralize the results Blea &1- gebras f prjectjs i lcally cvex sp8~e~ R~ ~~t f~~!~ [lj ~~d [25]. It will be cveiet t give five seemigly differet cditis, which tur ut t be equivalet.

23 16 Lemma ~ A E..~!. semi-prime rig ~ I!! arbitrary ideal ~ A. ~ there exists.!. family (b) c I ~ ~ ~ fllwig cditis ~ satisfied; (i) ~ family (b) ~ mutually disjit: b\l\l = 0, if \I :; lj, (ii) ~ family (b)~!!.2itrivial: ICC = (~b\la)cc. Prf. Csider the cllecti f all subsets f I, which satisfy the cditi that ay tw distict elemets have prduct O. A applicati f Zr I s lemma applies t the effect that there exists a maximal subset ~ = (b) havig this prperty. 'We claim that the family ~ als satisfies (ii). Up lettig I\I = b\la, we i:.ave t prve that ICC = (EI)cc. Sice t shw that I cc c ( )cc ( c)c ~I\I C I, we ly eed ~I\I = r\l r, equivaletamuts t Iccr c ={ Ol, Which, by Lemma 2.1.5, c c ly, I ::> I\I i the latter \I c i tur is equivalet t I"I" = {Ol. Nw csider ay b i Illr c c c :r~r every lj we ha.ve boilj c IlJ"I" C IlJl'tIlJ = {Ol. Hece, b e: I aihilates all members f ~, ad s b = O. " Fr a mre cci~~ furmulati f +'he ext therem we shall adpt the fllwig termilgy: a. family f ideals [rig elemets] is said t be disjit if ay tw distict pair has zer itersecti [prduct]. Furthermre, we shall say that a itersecti la f ideals is cutably accessible if there is a cutable subfamily f idices (~) fr which OI a = r a Cl we w are ale t Cieri ve the fllwig result.

24 'lberem ~ A ~ ~ cmmutative semi-prime rig. ~. the fllwig assertis ~ equivalet. 17 (i) (ii) ~ disjit family!. arbitrary -zer ideals.!!. cutable; ~ disjit family!. arbitrary -zer regular ideals..!!. cutable; (iii) Ay disjit family!. -zer elemets!!!. A!.!!. cutable; (iv)!h!:. itersecti!. ay decreasig family!. regular ideals is cutably accessible; (v) ~ itersecti 2!.!!!. arbitrag family.2!. regular ideals is cuta-.!l. accessible. Prf. We will shw (i) => (ii) => (iii) => (iv) => (v) => (i). (i) => (ii). Trivial. (ii) => (iii). Let ~ = (b a ) be a family f -zer elemets i A such that bsb a = a fr S ~ a. It is t be sw that 11 is at mst ~utable. Csider cc cc cc cc the ideals I a = baa. By Lemma we have I a Is = (IaI ) S = {a} = = {a} fr B :; a. Thus ~ is at mst cutable, sice by (ii), the family (I a ) is s. (iii) => (iv). Let ~ = (I a ) be a family f decreasig regular ideals. We have t prve that there exists a cutable subfamily (1'11) such that ra = I Cl Csider the ideal I = ~. By the previus lea there exist elemets b v c I such that (a) bvb u = 0 (b) ICC = fr v:;u, By (iii) the family (b v ) is at mst cutable, say (b ). Sice the family (I ) is decreasig, a the f~.ii:." is icreasig, s we may assume that fr every, b e: I~. fr s:me CX

25 18 Thus Eb A (EbA)CC c (EI~)cc = (I~)c. It fllws that Ia ::> '!his prves the asserti. ra We have trivially that I~:::> Ia. The implicati (iv) => (v) fllws frm the ext geeral resuit, which has scme iterest f its w. Lemma ~ X ~!!:. pit ~ ~ C a cllecti f subsets f X which is stable uder cutable itersectis ad which ~ the fllwig prperty: ~ itersecti!?! ~ decreasig family ~ C ~ cutably accessible. ~, every itersecti.2! members f C ~ cutably accessible. Prf. Give ay subcllecti r f C, we must exhibit a cutable subset ~ ff, such that > = 1: Let'~ be the cllecti f all fiite itersectis f members f 1. Csider the family f the cutable subsets ~,-;;/ f f We shall write t 1 '" ~ wheever ~ = ~ It is easy t veri fy' that this des defie a equivalece relati. Dete the class ctaiig - ~ by ~. We w defie a partial rder i the set f these equivalece - - classes: 4>1 > ~2 if fr represetatis we have ~l c ~2. Agai it is readily verified, that this relati defies a partial rder. Next, let (~) v be a descedig family ad write ~ v = stahle uder cutable itersectis, ea~~ {Lv ' e:, Lv belgs t C., N}. Sice C is ~iil(;e the ra.wll;y '*'.I.::> c.lt:::>cl::ildig, there exists, by assumpt;i, a cutav ble subset {~v } C {~v} 9 such that The right-had side features a.itersecti f cut!:'_bly!!'.!!...'!y!!'.<>mbe!"s f fa', Let 4> be the set f these elemets.

26 - - te, clearly, ~ is greater tha ~", fr each ". Hece, by Zr's lemma, there exists a maximal equivalece cless 41 max Claim: Suppse t, the, sice fl, L t max } certaily ctais fl, L fi, - 19 there exists a elemet L 1such that f1{l, L e: cfl }L::;' fl, L e: ~max}. max 0 Let ~ = {LL ' L 4l max }, the t is a cutable subset f ~ fr which l > ~max ad $0 # ~max This vilates the maximallty f i max, whece the statemet. I rder t show' the implicati (iv) => (v) we eed ly t remark that the set f regular ideals is stable uder cutable itersectia. (We eve kw that it is clsed uder arbitrbl~ itersectis.) (v) => (i). Let ~ = {I a } be a family f arbitrary -zer ideals satisfyig ISl a = {OJ fr B ::;. a. It is t be shw that this family is cutable. Csider the family f regular ideals {I~}. c c By (v) there exists a cutable set {I a } such that l a = Ia. Claim: cfl = {I a }. If t, cfl wuld ctai I with I # I~ fr all. The, IIa = {OJ ad s c c c c I c: I fr all, whece I c l = OI CX a a Hece I c I ad 60 Ie = {OJ i the semi-prime rig A. This prves the asserti. Remark 1. As the prf shws, the therem remais valid if everywhere the the expressi "cutable:' is replaced by "f cardiality \-\", where '>..1 \. r,,,;, f'i. Remark 2. Cditi (iii) eables us t ccmpare ur results with results f varlus authrs [1] ad [25]. Remark 3. Cditi (v.) will freq,1.letl:,' be used i this sequel.

27 20 We are OW' ready' t defie the cutable chai cditi. Defiiti ~ camutative semi-prime rig ~\tisfies the cutable chai cditi (c.c.c.) if it satisfies e f the five cditis f Crllary ~ X ~~ cmpletely regular tplgical space ~ A ~ algebra! ~ buded cmplex-valued fuctis E!!. X.!!!!;. fllwig assertis ~ equivalet: (i) (ii) A satisfies ~ cutable chai cditi; Every disjit family.2!. -empty pe ~.:i!!. X k ctmtable; (iii) Every family 2.!. ~ subsets (Oa)! X ctais!. cotmtable ~ family (0Cl) ~ ~ UO~ ~ dese!e. UOa Prf. We vill show' (i) <=> (ii), (i) <=> (iii). (i) => (ii). Let (Oa) be a family f mutually disjit pe subsets f X. The, sice X is cmpletely regular, there exists fr ea.ch a a buded ctiuus fucti fa such that fa "I 0 ad fa = 0 ff 0a. By the cutable chai cditi fr A, the family (fa) is at mst cutable ad s is the family (Oa). (ii) => (1). Let (fa) be a disjit family f -zer fuctis i A. We viii shw tha.t (fa) is cutable. Csider the family f the pe sets 0a = {xex, fa(x) :# Ol. The, S :# (l implies 0allOS is empty, '",hece the result.

28 21 (i) => (iii). Let (0 ) be a family f pe subsets f X. a Csider the set f ideals {Ia; I a = {fea, f = 0 0a. The, by Lemma 2.2.2, I a is regular fr each a. O accut f the previus therem item (v), there exists a cutable family (I a ) such that I~ = Ia. Whece, {fea, f ". 0 UO }= {fea, f = 0 UO a }. If UO a were t dese i UO a, there wuld exist a pit X E UO a ad a pe eighburhd U f X Sice X is capletely regular such that (UO a )f\u = ~ ad X E UUO a there exists a fucti f fo(x O ) :I 0 ad f = 0 utside f UflUO a Thus ctradicti. E A such that f E m~ ad fl m a, a (iii) => (i). Let (Ia ) be a arbitrary family f reg.ular ideals. We will shw that there exists a cutable subfamily (I a ) such that ra = Ia By Lemma 2.2.2, we kw that fr every a there exists a pe subset 0a f X such that I a = {fea, f = 0 0a}. The ra = Q{f A, f = 0 0a} = {fea, f = 0 U0ell. But there exists a cutable subfamily (Oa ) such that UOa UOa. Hece ra = {fea, f = 0 UO~} = r a is dese i Crary ~ X ~!. cmpleteq regular tplgical space which satisfies ~ cutable chai cditi ~ ~ be!!!!. pe subset f X. ~ there exists!. cutable icreasi[ family.2! pe ~ (O) ~

29 22 Prf. Sice X is cmpletely regular ad 0 a pe subset f X, there exists fr each x e: 0 a buded -egative ctiuus fucti f such that f(x) 'I 0 ad f = 0 ff O. Hece the set 0 ca be writte as = U{xe:X, f(x) # O}, f where the ui is take ver all buded -egative ctiuus ruetis f, which vaish utside f O. By the previus crllary there exists a cutable subfamily (f ) such that ~{xe:x, f(x) # O} is dese i O. Withut lss f geerality we may assume that 0 ~ f(x) ~ 1 fr all x ad all. Defie f(x) ~ E 2 f(x), the f is buded, ctiuus =l CD - ad -egative. Mrever ~{xe:x, Fially let O = {xe:x, f(x) > - l }. f(x) 'I O} = {xe:x, f(x) # O}.

30 23 CHAPTER III SCME CCMMENTS ON STRICTLY POSITIVE F1JNCTIONALS 1. Prelimiary remarks. This chapter is etirely devted t a existece prblem psitive measures. Let A be a cmmutative C*-algebra. A psitive fuctial ~ E: At = (A, 11 11)' is said t be strictly psitive if f E: A, f:;' 0 implies <f*f,~> :;. O. Des A pssess a strictly psitive fuctial? EquivaletJ.y, let X be a capact Hausdrff space. Des there exist a strictly psitive prbability measure, i.e. a regular psitive Brel measure lj such that lj( X) = I ad such that, fr every -vid pe set 0, we have lj(o) > 07 affirmative. There are a few well-kw cases fr which the aswer is ~ = 2-r r h. First. if X is separable, we may take l = =l u' where u 1S t e p01t th evaluati at the elemet f a dese sequece i X. Secd. if X is the clsure f a ~ subset f a capact grup, e may take the Haar measure; see e.g. [13]. Chapter XI. Let A be a cmmutative C*-algebra. '!he the existece f a strictly psitive fuctial ~ E: At implies that A satisfies the cut~ ble chai cditi. Let {f y y [; f} be a famij.y f psitive elemets i A fr which 1/ fyli = I fr all y E: r ad f f = O. wheever Y1 :;. Y2. We will Yl Y2 shw that r is cutable. Let 0 be ay psitive umber ad csider the set

31 24 Claim: r is fiite. I f'act. if t. the r wuld ctai at least cutably may distict elemets Y1' Y 2 '!he sequece {gk}. defied by wuld have the prperties: Jlgkll gk = ri~lf y. ' 1 ken = 1 fr every k ad <~,~> = ri~l<fy.,~> ~ ko. 1 We may suppse that 114>/1 = I, whece fr all k, which is impssible. Hece, r is fiite ad thus r = U{rl.' = 1. 2, } is cutable, ideed. " We als have the fllwig easy prpsiti. Prpsiti ~ 4t ~.! psitive f\mctial ~ ~ coolll1utative C -algebra A. ~ fllwig assertis ~ equivalet: (i) ~ fuctial 41!!. strictly psitive; (ii).e2!. every -zer ideal I ie. A, da,4» 'I {Ole Prf. (i) => (ii). Let a # f E I, the f*f E IA ad <f*f,41> 'I a. (ii) => (i). Let f E: A, f 'I a. Csider the ideal I = fa. The <IA,~> 'I {a}, ~ there exists a elemet h E: A such that <fb,4» 'I a ad s by the Schwartz iequality: whece <f*f,~> 'I a. a 'I l<fb,~>12 ~ <f*f,4»<h*h,4», It, therefre. seems atural t csider ideals f t'.h". f,,!'!!! 14> = {fea, <fg,4» = a fr all gea}, where </l is ay elemet f A'

32 25 Ntice that if A has a idetity, the I~ is the"larsest"ideal i the kerel f ~. The the task will becme t prve the existece f fuctials ~ fr which I~ a {Ole As pited ut abve, it is ecessary t impse the c.c.c. A. This, hwever, des t seem t be sufficiet. The reas is that the c.c.c. essetially says smethig abut regular ideals: I I... {OJ implies Icclcc = {OJ, r i terms f pe sets O ZlI ~ implies It{O )I)It{O ) ~, where I, I are arbitrary ideals ad 0, 0 are arbitrary pe sets, respectively Regular fuctials ad rmed algebras. I this secti we will csider a tpl~ical algebra which is cmmutative ad semi-prime. Mrever we will assume that fr every ideal I c: A the "prjecti mappig" p: IA + I C A -+ la, defied by c p(a + b) a, a la, b I A, is ctiuus. Remark that a C*-algebra satisfies all these cditis. By A' we will mea the ttality f all ctiuus fuctial~ defied A. We will say that a fuctial ~ A' is regular if I~ is regular"(~. I~C = I~). Example. Let A C[O,I], equipped with the supremum rm ad g E A. The the fuctial f -.. f1f(t)g(t)dt is regular. Oe f the prblems we face will be whether r t there exist regular fuctials. The fllwig lemma gives sufficiet cditis i rder that the regular fuctials separate the pits f A.

33 Lemma ~ lli. tplgical ccmmutative semi-prime algebra A satis-.!z. ~ f11evig cditis: ( i ) ~ tplgy!! lcally cvex; 26 (if)!2!. every ideal I, ~ mappig defied El. p: IA + ICA -+ A. ~ ctiuus; p: a + b -+ a, (iii)!2!:. every regular ideal I O ' I ~ {O}. there exists!. fuctial <fl ~ ~ <ra.cf! 0> ~ {a}.!2!:. every clsed ideal I.!2!: which ICC = I ~ MI<fl., regular}.. {a}. Prf. Let 1 0 = {I<fl' ~ regular}. By Therem = Ig c We first cc cc ~ prve that ~ e A' implies IljI ~ 1 0, Suppse t. i:.!.. assume 11/1 Al T I fr sme!p. Csider the fuctial defied by W : a + b -- <b.lji> The lji is ctiuus its dmai. Let lji0 be a Hah-Baach extesi f li t all f A. The clearly I~c C IlJi. Fr the cverse cclusi we have by deiti 11JJ = {xea. <xy.tp > = a fr all yea} c {x A. <xb.~> =a fr all bci~} fer- _"... Al u...&...j". J ""'.M.J

34 27 We cclude that I~c = I w Our abve idirect assumpti w becaes = { cc 1 cc J 1 0 I~, I~ = I~ = II w = II~ T la, which is impssible. Csider w the ideal I", {'II, where ~ is a ele '/'0 0 met i A' fr which <la,eil > :I {l hlds fr every clsed ideal I c: A with ICC = 1 0 'Ihe, by the defiiti f I~ O <( I ell ('\1 )A,~ > = {}. 0 a, we have the ther had, by the prperty f 41 0 ad assumig that 1 0 :I {O}, we have Thus, I a = {l. «1cjlO~10)A,$O> :I {Ol. The prf f the ext lemma is rather techical. Lemma ~ ~ tplgy!2!: A ~ defied z.!!:.~. ~ agai A be semi-prime, cmmutative ~ M (il).2f. ~ previus lemma!:. satisfied. ~ (41 ) ~!!:. cuts.ble family 2! regular f'uctials i A'. ~ there exists a elemet 41 e: A' such that I", = rrr"' If' " Prf. We will cstruct a sequece f regular fuctials (lfi ) such that fr all : (i) IItJ1 li < 2, "' (if ) I = I, ms 4l (iii) <b,tjl > :: <b,lfi + > fr all b l e: 1$ A. We will assume that 1/41 ll < 1 fr all. The cstructi emplys by iducti. First, let WI = 4>1. Nv let the fuctials lfi 1,,tJl be cstructed i such a way that (a) Ii wkll ~ 2 - e:, k = 1,, 1 > e: > 0, (b) I - T all k ~, Wk - lsk"'$1' (c) <b,1jk> = <b,w l >, ~ k ~ I, b e: I~lA.

35 We will cstruct a regular f'lmctial W + l such that IIw + l ll ~ 2-2-~ 28 ad the family 1jIl ' 'W + l satisfies (b) ad (c) with replaced by +l. By (ii), there exists a cstat c such that, fr all a e: b e: 1$ A, the iequality ~ Cz1ila + bji is valid. Defie I1jI A ad by where a e: Ill! A, b e: Ie; A. 1JI The 1<80 + b'~+l>1 = 1.. <a,4l + l > + <a + b,lji>1 2c 5 e: /I all + (2 - e:) 1/80 + b /1 2c ~. 1/80 + b II + (2 - e:) lia + b 1/ 2c = ( e:) bll. Let lp+l be a Hah-Baach extesi f 'W+l t all f A, s that fr all x e: A. I<x,tP+l> I < ( e:) II x 11 The, the family lp l,,lp+l satisfies (c). Let us prve (b); the lp + l is autmatically regular. By defiiti Ill, = {ae:a: <AX_tlI.'> = 0 f')'!' "" '+1.,,~A\... J LJ'T'.L --- C {ae:a, <aby,lp > = a fr all be:i~, all ye:a} c = {ae:a, C 1lJJ } aiw c {ae:a, c = ai {a}} ID C 1 1" I1I~1 = ' ' = ICC 1jJ = I lp

36 29 Hece ItSJ :: 1tP 1"41 +l +l (by defiiti) = {ae:!1p, <8X,l/I+l> = 0 fr all x A} (defiiti f ~+l) = {a. l", <ax'~+l> = 0 fr all XE:A} = 1l/I1~ +l (iducti hypthesis) c 1~ () tu~ I'lI~ 1 +l The sequece (tp ), btaied i this w~, clearly satisfies the fllwig cditis (i) II ~ II < 2 fr all, (H) lip c I... fr m ~, m "' (Hi) <b,1/im> :: <b"> fr m > ad b r~ A, (iv) llji :: r, Fially, let ~ use iducti. First, we <8X,~ > = 0, fr all x = E.. 2- ". We Dcl 'l' claim that I~ = l",. We shall agai prve that I. C ItSJ If a E: I~, the 1 0 A, s certaily <aby,4'> = 0 fr all b I~1 ' all y A. But, by the prperties f the sequece ("'), we have ad s whece We ext cc a I,I. = I1P '1'1 1 shcrw that I. c. 1.,. imr>hp.5 I.L 'PO "' '+'0

37 30 By defiiti, we have (by the fact that (by defiiti f 14> = {a. Itjl' = {8 I1fI, <ax'~> = 0 fr all xe:a} Ek~12-k<ax,~> = 0 fr all xe:a} I"L.. C I, c c:i,) 'I'Il -l 1 = {8 I 1fI, Ek=~+12-k<ax,tI'k> = 0 fr all xe:a} {8 I1jJ' Ek=~+12-k<abX,1I1t>=O fr all be:i~+l' all xe:a} 1/J ) = {ae:i1jj, = {8 I1fI ' = {8 I1jJ, = I ICC 1jJ: 1fI+ l fr all be:i~, all xe:a} +l {}} It fllws that r.l. C r1jj The reverse iclusi ltji C I~ fllws '+'0 0 directly frem the defiitis. Therem !;!. ~ tplgical cmmutative semi-prime algebra A satis-!1.. ~ fllwig cditis: (i) ~ tplgy ill defied ~!!. ~; (ii)!:!:. every ideal I, ~ mappig p: IA + rca -+ A defied ~ c. p: a + b -+ a, vhere a e: la, b e: I A, ~ ctiuus; (iii) ~ algebra A satisfies ~ cutable chai cditi.!h!:!:.. ~ fllvim assertis ~ equivalet:.. CPa E }\.- ~ ~ <IAt~> " {J f2!:. every clsed ideal I f2!:. which ICC = 1 0, (b) There exists a fuctial 1jJ ~ ~ 11jJ = {O}. 0

38 Remark. The wrd tplgical m8\y be emitted if i asserti (a) "every clsed ideal I fr which ICC = I " is stregtheed t "every ideal I fr which ICC = I ". Prf. (b) => (a). Let 1 0 be a -zer regular ideal (that is, I~c = 1 0 # {O}) ad let ICC = 1 0, The <IA,~> # {Ole I fact, if 31 <IA,tJi > = {O}, the I C ItJi = {O} ad s {O} = 0 (a) => (b). By Lemma we have {a} = fl{i4>' ICC = I I~C = 14>}. Frem the c. c. c., we ifer that the itersecti is cutably accessible ad s there exists a cutable family (4)) f regular fuctials such that {a} = I4>' But, by Lemma 3.2.2, we kw that there exists a fuctial ~ 0 E: A' such that IljI = I4> 3. Strictly psitive fuctials. I this secti we shall apply the precedig results t a cmmutative C*-algebra. We fllw stadard termilgy i callig a elemet lji A' hermitia if the fuctial lji*: x -+ <X4,w> cicides with lji r, what is equivalet, lji takes real values the hermitia elemets f A. A fuctial ~ E: A' is called psitive if it takes egative values the psitive elemets i A. It is well-kw that a psitive fuctial is hermitia. Every cf> i A' ca be writte i the frm ~ = 1jI + i ~, where ~ ad ~ are hermiti a: simply let I 2 I 2 WI = (cf> + <P*)/2 ad W 2 = (4) - 4>*)/2L We als kw that every hermitia W A' admits f a Jrda decmpsiti ~ = WI - ~2' where WI ad lji2 are psitive fuctials i AI ad jlljlll ::I IiwIIi + "'1'2 11 (See [7](2.6.4); accrdig t Grthedieck: this deco!!!psiti!! whether r t A is cmmutative).

39 32 It fllws that ay tfl e: A' ca be Wliquely represeted i the frm 4 tfl = E=li tfl, with ~l '~2 ' tfl 3 ad tfl 4 all psitive. The ctets f the ext lemma is that fr suitable chse psitive fuctials ~l' ~2' '3 ad '4 fr which, = E;li~ ~ we have lei> = 1' It the easily fllws that fr we have 4 '0 = 1:=l' 14>0 = 14>. We als 0 eed the fact that, fr ay tw pasi tive elemets a ad a. i A, I 2 we have {he:a, 0 5 h s a l + a 2 } = {he:a, 0 5 h 5 all + {he:a, 0 s h 5 a 2 }. It the fllws that fr ~ ay hermitia fuctial the mappig a -- s up{ <h, ljj>, 0 5 h 5 a} is liear the ce f the psitive elemets i A. Fr mre details vectr lattices see~. [23]. Lemma ~ A ~ ~ cmmutative C*-algebra ~ let e: AI. The there exists a psitive fuctial ~ e: AI such that I... = I 'f 'f0 Prf. We first prve that, if a e: A ad 0 :s h 5 aa*, the h belgs t the clsure f aa. Sice A is a cmmutative C*-algebra, we kw that a clsed ideal I is the itersecti f the maximal ideals ctaiig I. It fllws that, if ~ detes the maximal ideal space f A, the ideal aa is dese i {Ker, ee:~, <a,e> = a}. S, if 0 S h ~ a.a*~ the <a~e> = 0 implies <h:> = O. Hece h bel!!es t the clsure f aa.

40 33 Next we write ~ = ';:1 ;. i1ll2 t where ~1 ad $2 are hermitia fuctials. By defiiti we ha e I~ = {ae:a, <ax,~> = 0 fr all xea} = {ae:a, <ax,~ + illl> = 0 fr all xea} 1 2 C {aea, <aa*x,lli + i~ > = 0 fr all xe:a, x = x*} 1 2 = {aea, <aa~x,$ > = 0 fr all xea, x = x*} 1 (sice the hermitia elemets spa A) 1\ {ae:a, <aa*x, $ > = 0 fr all %e:a, x = x*} 2 (sice 1111 f"ii, = {ae:a, <aa*x,~ > = 0 fr all xea} 1 l\{aea, <aa*x,~ > = 0 fr all xea} 2 = {ae:a, aa e: I~ I'\I~ } 1 2 is clsed) 1 2 = 1$ f"itij 1 2 The reverse iclusi I~ f'\itji c 14l is trivial, whece I~ = IljJ 1"101, ~2 Nv let 111 be a hermitia fuctial. Defie its "psieive variati" tp ' l First fr psitive elemets i A: <a,ljj > = sup {<h,1jj>, 0 5h 5 a}, a ~ O. 1 (!;,:.. see [23], p.2ll) Sice A is a vectr lattice,~ 1 is liear A+, the ce f the psitive elemets. Fr arbitrary a E A, write a = I:~li~, where a i is psitive fr i = 1, 2, 3, 4 ad a 1 a 3 = a 2 a 4 = O. Defie <80,111 1 > by liear extesi. The.pI is a psitive ctiuus fuctial A. Let 1jJ2 = 1jJ1-1jJ. The fr every elemet a e: A+ we have ~d..;)u ~ is psitive. 2

41 34 Next we prve that I, z I'l~W2. Let a I,; that is <&A,,> = {a}. If a ~ h ~ aa*, the, by the reasig at the begiig f' the prf, h belgs t the clsure ~ the ideal aa. It fllws, by ctiuity, that <h,1/!> = O. Hece <&&*,1/11> = sup{<h,t/1>, ash S &&*} = O. Sice '1 is psitive, we cclude a : I, Similarly we may shw that a I, 1 2 Hece I, = I~{'lI~2' the reverse iclusi, I!ppI llil C I,t beig trivial. This methd ca be emplyed fr the hermitia fuctials!p l ad "'2 i 4> =, + HI t prvidig us with fur psitive fuctials ~1' ~ t '3 ad ~It t s that I, = OI lfl We w write dw a result which is similar t Therem Therem ~ A ~!:!; cc:mmutative C*-algebra. ~ fllwig assertis ~ equivalet: (i) (ii) There exists!. strictly psitive fuctial ~ A'; There exists!. mappig T: A -- A', which.!!. ~-l2:"e.!2!:. which <ab,tc> = <b,tea>.!2!. ill a, b, c!e. A; mrever (i!. A ~ ~ pssess a idetity) A satisfies ~ e.e.c.; (iii)!:.2!: every regular ideal I ' 1 0 :; {a}, there exists!. fuctial ~0 i2. A' such that <I,~ > :; {a}, fr a.ll clsed ideals I fr which ICC = 1 0 ; mrever A satisfies the c.c.c. Prf. (i) => (ii). Let ~ A' be strictly ps~tive. Defie T: A -+ A' as fllws: if a : A, the Ta is the fuctial which assigs t x the umber <ax,4». Thus <x,ta> = <ax,et» fr all at x : A. It is readily verified that <ab,te> = <b,t(ca» fr all a, b, e i A. We shw that T is e-t-e. If a is a elemet f A fr which Ta = 0, the <x,ta> = a r <ax.et» = a fr ~ll A ~ A. Iu particular, <aa~,~> = 0 ad s aa- = 0, r a = O.

42 (ii) :> (i). Let T: A -+ At be as i (ii). Defie. fr every a E A. the fuctial "'a A by <x.el>a> = <x.ta>. It is a matter f rutie t verify that the ideal Iel> is equal t a I", 2 {x A, ax =J. a Hece, by Prpsiti 2.1.4, I~c = I~ a a If A has idetity, we kw, by the c.c.c., that the itersecti {Iel>. a A} is cutably accessible ad hece there exists by Lemma a a fuctial el> such that Icjl = Uel>a' aea}. Fra the fact that Icjla = {xea, ax = a}. we see that Iel> = {Ole By the previus lemma we 35 may assume withut lss f geerality that el> is psitive ad s I~ shwig that el> = {xea. <x*x.el>> = a} = {OJ, is strictly psitive. If A des have a idetity e, the fuctial ~e: x -+ <x.te> has the prperty Iel> = {a l. Agai we may assume that el>e is psitive ad it fllws e that cjle is strictly psitive. (i) => (iii). Let cjl be a strictly psitive fuctial A. The. <I.Ijl> ~ {J fr every -zer ideal I. Hece, if ICC = I where ICC = I # {a}, the <I,cjl> ~ {Ole 0 That A satisfies the c. c. c. has already bee prved abve. (iii) => (i). This is a straightfrward applicati f 'Iherem ad the previus lemma. Remark 1. If cjl At is a strictly psitive fuctial, the the mappig: {a,b} - <ab*.<1>0>, defied A x A, is a ier prduct which makes A it a. Hilbert algebru; see [6], p.330.

43 Remark 2. A smewhat weaker frm f (iii) is sufficiet t cclude (i). Fr every -zer regular ideal I C A tgether with ay cllecti cc. c:. {I, I = I } f clsed ideals with the prperty that every cutable O (ad every fiite) itersecti {I' = I, 2,.. l, I e: '. belgs t, there exi~ts a fuctial ~ e: Al such that <I,~> ~ (OJ fr all ideals I e: ; mrever A satisfies the c. c. c. Clsely related t this remark is the prblem at the ed f this chapter. I the light f Therem 3.3.2(iii), the existece prblem fr strictly psitive fuctials ca be reduced t the fllwig e. Let I be a-zer regular ideal!.:.!;.. I~c = 1 0 ~ {Ole As i (iii), we csider the cllecti f thse clsed ideals I fr which ICC = I Nw select i every such ideal I :f= {xil. a -zer psitive elemet XI ad lk at the family Claim. 'Ibere exists a strictly psitive fuctial A if ad ly if the family f ca be chse i such a maer that there is a psitive fuctial cp e: A' which des t vaish at a:y pit f 1..--r I fact. suppse, idirectly, that fr each chice f "1 every psitive cp e: Al the set ~ = {x e: ~ <x,cp> = O} is -empty. Sice, fr ex> - every sequece ( cp ) ) C A', ~ ~ O. li l i 1, the fuct~al cp = L: =12 cp has agai these prperties, it fllw's that fr ay cutable cllecti f-~ -... ~s- ( f ), cp the itersecti jcp (= J ) is -vid. 'Ibis is impssible if cp A = C(X), where X is cmpact ad separable. Neither is it pssible i case 1-is weakly cmpact (= weakly cutabj.,..v capact accrdi!lg t 'C"'\...., ~. r.-. -., 1><5 'J ~Ver.Le... l1, st:t:~. lcjj, p. u Hwever, we...ere t able t cstruct such a weakly cmpact family r.

44 37 As ather csequece f the therem we have Therem ~ X ~ A cmpact Hausdrff space. ~ fllwig assertis ~ equivalet: (i) There exists.!. strictly psitive fiite Brel measure 1.1 ~ X; (li}!h!t space X satisfies ~ c.c.c. ~,!2!: ay -vid regular pe ~ 0, there exists.!. buded regular psitive measure lj.sm. 00 such that )J (O) > 0 fr every,2e.e.a. set 0 which is dese i Prf. We csider the algebra A ~ C(X} f all ctiuus ccmplex-valued fuctis X. Recall the e-t-e crrespdece betwee regular ideals i A ad regular pe sets i X; see Therem The mappig I -+ U {xex, f(x} ~ O} fei is a bijecti betwee the cllecti f clsed ideals I A ad the tplgy f X: the cllecti f the pe subsets. Its iverse is give by 0-+ {fea, {XEX, f(x} i- O} c e}, where e is ay pe subset f X. The restricti f these mappigs t the regular ideals ad regular pe sets respectively establishes a e-t-e crrespdece betwee the regular ideals ad the regular pe sets. Csider a pair (I, 0 ), where e is a regular pe set belgig t the 0 0 regular ideal I The uder the abve mappigs the cllecti f ideals {I A I 1 d I CC -- I } is i t d'th h c::, c se, e- -e crresp ece V1. t e cl- O lecti f pe sets {a c X, a pe ad dese i a~}. v After these preparatry remarks we v prceed vith the prf f the theerem.

45 38 (i) => (ii). Clear. (ii) => (i). By virtue f Therem it is sufficiet t exhibit a fuctial satisfyig cditi (iii) i that therem. Let 1 0 be ay regular ideal i A = C( x) ad 0 0 its crrespdig regular pe set. By (ii) there exists a measure lj pe subset 0 f 0 Give ay set = U {XEX, f(x) ~ } 0 such that lj (0) > 0 fr every dese 0 clsed ideal I fr which ICC = 1 0, the fei is pe ad dese i 0 0 Sice lj is regular there exists a cmpact subset K C 0 such that lj (K) > O. ~t f be ay fucti satisfyig the 0 fllevig cditis: f(x) ~ 0 fr all XEX, f(x) = 1 fr all xek ad f (x) = 0 fr all x ff O. Such a fucti exists, sice X is cmpact (ad s rmal). The fucti f 0 belgs t I ad we Sice lj may be viewed as a ctiuus fuctial a atural way a subspace f C(O), the measure lj 0 0 have lj (f ) ~ O. 0 C(O ) ad I 0 is i defies a ctiuus fu::tial I Let cjl be ay Hah-Baach extesi Oi.' lj t 0 0 all f A, the ~ des satisfy cditi (iii) i Therem Agai, let X be a cmpact Hausdrff space, Take a -empty regular pe set 0 0 i X ad csider the fllwig hypthesis 0 0 Hypthesis (*). There exists a family fur' YEf} i A = c(x) & tgether with a family f pits {xy' YEf} c X, such that the fllwig cditis are satisfied: (ii ) r'\'p"'l...r-... set 0 there exists a elemet y r such that {XEX, u,/x) ~ O} CO;

46 39 (iii) The fuctial ~: ri~l~iuyi -+ ri~iaiuyi(xyi) is well-defied ad ctiuus the vectr space spaed by the family {u, YEf}. Y Remark 1. '!be clsure f {xy, YEf} has -vid iterir. Remark 2. Whereas the cllectis {Uy} ad {xy} ca always be chse i such a way that (i) ad (ii) are satisfied, (iii) is the crucial cditi. Remark 3. A mtive fr lkig at this type f cditis is furished by the fact that if istead f the family {Uy, ye:f} we wuld have take the cllecti f characteristic fuctis the the fuctial {Xc' pe ad deae i 0}' ri=lai XCi -.. 1: i =lai has prperty (iii), if we take the supremum fr defiig the tplgy. Therem ~ X ~!:. ccmpact Hausdrff space.! sufficiet cditi!2!.lli. existece 2!. ~ strictly psitive measure!!.~ every -vid regular pe set satisfies hypthesis (*) ad that X satisfies the cutabl~ chai cditi. Mrever, i!. X i!. cected ~ these cditis ~~ ecessary. Prf. (sufficiecy) We will check asserti (iii) i Therem Let I O be ay regular ideal ad 00 be the crrespdig regular pe set. If {Uy, ye:f} ad {Xy' ye:f} are as i hypthesis (*) fr the set 00' the the fuctial 4>0' which is defied the vectr space spaed by the family {Uy, ~~r} ~~its a H~~-Bwiacll extesi t all r c(x). 1bis extesi satisfies the ~ditis put frth i Therem item (iii).