DISJUNCTIVE LINEAR OPERATORS AND PARTIAL MULTIPLICATIONS IN RIESZ SPACES
0000 Promotor: dr.B . van Rootselaar, hoogleraar in de wiskunde YY/ï Gto \ ^So c
B. VAN PUTTEN
DISJUNCTIVE LINEAR OPERATORS AND PARTIAL MULTIPLICATIONS IN RIESZSPACE S
PROEFSCHRIFT TER VERKRIJGING VAND EGRAA DVA N DOCTOR IND ELANDBOUWWETENSCHAPPEN , OP GEZAGVA ND ERECTO R MAGNIFICUS DR. H.C.VA NDE RPLAS , HOOGLERAAR IND EORGANISCH E SCHEIKUNDE, IN HET OPENBAART EVERDEDIGE N OP WOENSDAG 17DECEMBE R 1980 DES NAMIDDAGST EVIE RUU RI ND EAUL A VAND ELANDBOUWHOGESCHOO LT EWAGENINGEN .
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STELLINGEN
I Iedere semi-normp o pee n eindig dimensionale Archimedische Riesz ruimte E zodanig dat p(x)= p(|x| )voo r allex e Ei see nRies z semi-norm. De verwijzing die Aliprantise nBurkinsha w voor deze stelling geveni s niet terecht omdat deze stelling reedsi n196 1i sbeweze n door Bauer, Stoere nWitzgall .
Aliprantis, CD, enBurkinshaw ,0 . Locally solid Riesz spaces, AcademiePres s (1978). Bauer, F.L., Stoer,J .e nWitzgall ,C . Absolute and monotonie norms, Numer. Math.,Vol .3 ,pp . 257-264 (1961).
II Alsp ee n reguliere vectoriële pseudonorm: £ ^ IR is,N(p )={ xe £n; p(x)=0 } en R(p)= (p(x);x e %} ,da ni sdi mR(p )< n - di m N(p).
III Also pee n Archimedische Riesz algebra Eee n L-norm II-II gedefinieerd is dan bestaate ro pE ee nsemi-inproduct . AlsE ee n Archimedische $-algebrai se n(E ,II .II )ee n AL-ruimte,da ni s er een norm II-II 'o pE zodani gda t(E ,II -II')ee nHuber t ruimteis .
IV De lineaire ruimte B(X,Y)me t operatornorm van alle norm begrensde lineaire operatoren vanX naa r Y,waarbi jX * {0}e nY genormeerd e lineaire ruimten over F zijn,i sjuis t dan norm compleet alsY nor m compleet is. Het analogonva ndez e stelling inRies z ruimten,me t orde begrensd inplaat s van norm begrensde nDedekin d compleeti nplaat sva n norm compleet,i sgeldi g onderd ebeperkin g datd eord e dualeX vanX voldoe t aanX =£{0} .
Pfaffenberger, W.E. A converse to a completeness theorem, Amer. Math. Monthly, Vol. 87,no .3 ,p .21 6(1980) . Aliprantis, CD. On order properties of order bounded transformations, Canad.J .Math. ,Vol . 27,no .3 ,pp .666-678 . V AlsX ee n lineaire ruimte over K is,Y ee nDedekin d complete Riesz ruin en p:X - >Y eensublineair e operatorva nd evor m |Lx| (Lee nlineair e operator vanX naa r Y),da ni svoo r iederex e X d everzamelin g K ={ ye X ; p(x+ y )= p(x )+ p(y) }ee npre-kege l in X.
VI Zij voor allen e H den x n matri x A gegeven door A (i,j)= e ' 1J ' (i,j = 1 ,.. ., n). e+ 1 Voord espectraa lstraa l p(A )va n A geldt lim p(A)= _ •, .
VII Vekslere nGeile r hebben bewezenda tieder e voorwaardelijk lateraal com plete Archimedische Riesz ruimted eprojecti e eigenschap bezit. Bernau zegtva ndez e stellingee ngeneralisati e tegeve n intralie-geor dende groepen. Integenstellin g tot zijn bewering iszij n resultaat voor Riesz ruimten zwakker dan dat van Vekslere nGeiler .
Veksler, A.I.e nGeiler , V.A. Order and disjoint completeness of linear partially ordered spaces (Russisch), Sibirsk Mat. 1., Tom. 13,pp . 43-51, English transi.: Siberian Math.J. ,Vol . 13,pp . 30-35. Bernau, S.J. Lateral and Dedekind completion of archimedean lattice groups, J. London Math. Soc. (2),Vol . 12,pp .320-322 .
VIII De statistische selectie-e nrangschikkingstechnieken , waarvoor recent ind estatistisch e literatuur veel belangstelling aand eda g isgelegd , verdienen inhe tlandbouwkundi g onderzoek grote aandacht.
IX Het verdient aanbeveling deuitdrukkin g "een x-aantal" niett ebezige n zonder specificatie vanx .
X De overheid dientte n aanzienva nalternatiev e groeperingen geen alter natieve gedragslijn tevolgen .
Stellingen bij het proefschrift "Disjunctive linear operators andparti e multiplications inRies z spaces". B.va nPutte n Wageningen,1 7decembe r 19Î VOORWOORD
De aanleiding tot het schrijven van dit proefschrift was een projectvoor stel, gericht aan de Vaste Commissie voor de Wetenschapsbeoefening van de Landbouwhogeschool,da t als titel had "vectorial norms on linearspaces" . Ditwa s tevens de titel van mijn afstudeerverslag (mei 1976). Ditproject voorstel werd goedgekeurd en in september 1977wer d begonnen met de uit werking van dit project. Daartoe werden normen bestudeerd,di e waarden aannemen in Dedekind comple te Riesz ruimten.A l spoedig kreeg ik daarbij temake n met de operatoren, die in dit proefschrift een belangrijke plaats innemen en die zeer de moeitewaar d leken om teworde n bestudeerd. Voor verdere studie van heton derwerp was kennis van deze operatoren noodzakelijk en zo langzamerhand werden de ruimten die het bereik zijn van deze normen het domein van studie. Intussen bleek dat de operatoren die mijn belangstelling hadden ook bestu deerd werden door enkele anderen,meesta l vanuit verschillende achtergronden. Het resultaat is nu een proefschrift op het gebied van de zuiverewiskunde ; de titel ervan heeft met de titel van het projectvoorstel toch nog twee woorden gemeen.
Bij dezen dank ik mijn ouders voor hun zorg door alle jaren heen en ook mijn overige familie dank ik voor hun belangstelling. Professor Dr.A . van der Sluis dank ik vriendelijk voord e fijne afstudeer- periode en voor het vele dat ik in die tijd van hem heb geleerd. Mijn promotor,Professo r Dr.B . van Rootselaar,dan k ik hartelijk voor zijn hulp en zijn aanmoediging om de sprong in het oneindig dimensionale temaken . Ook de overige leden van de Vakgroep Wiskunde dank ik voor hun belangstel ling, in het bijzonder Drs.B.R . Damsté voor het corrigeren van de tekst in de zeer korte tijd die daarvoor nog overig was. Iwoul d like to thank Professor Dr.W.A.J . Luxemburg (Pasadena,California ) and Professor Dr.H.H . Schaefer (Tübingen,B.R.D. ) for their kind invitation to visit the conference about "Riesz spaces and order bounded operators", which was held in Oberwolfach, from June 24 till June 30,1979 . Jodien Houwers en Ans van der Lande-Heij dank ik voor het typen van dit proefschrift. Mijn vrouw Riet dank ik voorhaa r steun en voor het accepteren vanmij n dag en avondindeling. AMS(MOS) subject classifications (1980) 06F20,47B55 . CONTENTS page
Chapter I. RIESZSPACE S ] 1.Orde r relation 1 2. Partially ordered linear spaces and Riesz spaces 3 3. Elementary properties of Riesz spaces 6 4. Linearoperator s 13 Chapter II. SOMETYPE SO F CONVERGENCE 19 5. Sequences in Riesz spaces 19 6. Order convergence 19 7. Regulator convergence 21 8. Characteristic convergence 23 9. Comparison of convergences of sequences 25 10. Some notions of ideals 27 11. Continuity of linearoperator s 29 Chapter III.DISJUNCTIV E LINEAR OPERATORS 32 12. Orthomorphisms 32 13. Nullspaces of disjunctive linear operators 40 14. Examples of unbounded stabilizers 51 Chapter IV. f-ALGEBRAS 53 15. Introduction to f-algebras 53 16. Faithful f-algebras 57 17. Ideals and ring ideals 60 18. Ring homomorphisms inArchimedea n f-algebras 63 19. Normalizers and local multipliers on f-algebras 69 Chapter V. INVERSIONS AND SQUARE ROOTS INARCHIMEDEA N $-ALGEBRAS 72 20. Inversion inArchimedea n $-algebras 72 21. Square roots inArchimedea n $-algebras 76 22. A generalization ofth e Ellis-Phelps theorem 79 Chapter VI. PARTIAL f-MULTIPLICATION INARCHIMEDEA N RIESZSPACE S 81 23. Introduction and definitions 81 24. Intrinsically defined p<ï>-multipl icatio n onArchimedea n Riesz 88 spaces with a strong order unit 25. Orth(E)fo r Ea n Archimedean Riesz spacewit h a strong order 92 unit REFERENCES 95 SAMENVATTING 100 CURRICULUM VITAE 101 ChapterI RIESZSPACE S
Inthi s chapterw e give anexpositio n of theelement s of the theory of Rieszspace ;fo ra shorthistorica l introduction we refer to thebook s ofAlipranti s and Burkinshaw [1978] and Luxemburg and Zaanen [1971].
1.Orde r relation
An order relation on a non-empty set Si sa subset< of the Cartesian product Sx S o fS wit h the following properties: < istransitive ,i.e . if (x,y)e < and (y,z)e < for x,y,ze s,the n also (x,z)e < , < is reflexive,i.e . (x,x)e < for all xe S,< is anti-symmetric,i.e . (x,y)e < and (y,x)e < forx,y' eS implie s x= y .
Iti scommo n towrit e x< y instead of (x,y)e < .
A partially ordered set isno w defined as apai r (S,<)wher e Si sa non-empty setan d< isa norde r relation on S. Twoelement s xan dy are called comparable ifa t least one of thestate ments x< y ,y < xholds ,otherwis e theyar ecalle d incomparable. A non-empty subset Xo fa partially ordered set (S,<) iscalle d achai n in (S,<) ifal l pairs ofelement s of Xar ecomparable . (S,<) is said tob e totally ordered ifS itsel f isa chai n in (S,<). Ifx isa n element ofa partially ordered set (S,<) such that x< y for y s s implies x= y , then x iscalle d amaxima l element of (S,<). x isa minima l element of (S,<) ify < xfo ry eS implie sy =x .
Forelement s x,y,z ofa partially ordered set (S,<) the following nota tions are used: x< y for x< y &x # y; y > xfo r x< y; y > xfo r x< y; x< y< zfo rx < y &y < z; x,y< z for x< z& y < z; [x,y] for the order interval {zeS; x< z< y} . If X and Y are non-empty subsets of a partially ordered set (S,<), then X is majorized by Y, in formula X< Y, if x < y holds for all xeX and yeY. In that case Y is called minorized by X. If Y is a singleton {y}, then we write X< y instead of X< {y}, y is said to be a majorant of X then. Dually, if X is a singleton {x}, then we write x < Y instead of {x} < Y and x is said to be a minorant of Y. A non-empty subset X of (S,<) is called majorized in (S,<), in formula X <, if there exists a zes such that X< z, minorized in (S,<), in formula < X, if there exists a ye S such that y < X and bounded, in for mula < X< , if X is majorized and minorized at the same time. As a consequence, we have that a non-empty subset X is bounded if and only if there exist y,z e S such that X c [y,z] .
Now we can formulate Zorn's lemma, which we give in the following form.
1.1. Zorn's lemma. If every chain in a partially ordered set (S, <) is majorized in (S,<) then (S, < ) contains at least one maximal element.
An element xo fa partiall y ordered set(S,< )i scalle d asupremu mo fa non-empty subset Xo fS i fx i sa majoran t ofX an da tth esam e timea minorant ofth ese to fal l majorants ofX ,i nformul a X< xan di fX < y for somey eS the n x< y . Itfollow s from theanti-symmetr yo fth eorde r relation thata supremu m isunique . The supremum xo fX tf>+ is denote d bysu pX . Dually,ze s iscalle d aninfimu mo fX i fz i sa minoran t ofX an da tth e same timea majoran t ofth ese to fal l minorants ofX ,i nformul a z< X and ify < X fo rsom eye s theny < z.Als oa ninfimu m isuniqu e andi s denoted byin fX .
1.2. Definition. A lattice is a partially ordered set (S,<) with the property that for all x,y£S holds that sup {x.,y} and inf {x,y} exist in S.
In the sequel we write x.Vx^V... Vx for sup {x,,x2,...,x } and x,Ax2^...Ax for inf {x,,x2,..,x } . A lattice (S,<) is called distributive if for all x,y,zeS holds that (xVy)Az = (xAz)V(yAz). A lattice (S,<) is distributive if and only if for all x,y,z£S holds that (xAy)Vz = (xVz)A(yvz) (cf. e.g. Birkhoff [1967, I. 6 thm 9].
1.3. Definition. A lattice (S , <) is called (a) (order) complete if sup X and inf X exist in S for every non-empty subset X of S. (b) Dedekind complete if sup X exists for every majorized subset X of S and inf Y exists for every minorized subset Y of S. (c) Dedekind a-complete if sup X exists for every majorized countable subset X of S and inf Y exists for every minorized countable subset Y of S.
1.4. Definition. A distributive lattice ( Sj <) is called a Boolean algebra if 1 : = sup S and 0,: = inf S exist and if for every xeS there exists a (necessarily unique) x'e S such that xAx' = 0 and xVx' = 1. In that case x' is called the complement of x.
2. Partially ordered linear spaces and Riesz spaces
2.1. Definition, A tripel (EJ+J<) is called a (partially) ordered linear space if (a) (Ej+) is a linear space over H (b) (E, <) is a partially ordered set (cl) x < y implies x + z < y + z for all x^y^z e E (c2) 0 < x and A> 0 implies 0 < Ax for all Ae ]R and x e E.
In the sequel we abbreviate (E,+,< ) to E for fixed + and<. An element x of a partially ordered linear space E is called infinitely small with respect to 0 2.3. Proposition. A Riesz space E -is Archimedean if and only if nx < y /or x,y e E and all n e IN implies x = 0. Proof: ->- : follows directly from the definition +• : if nx < y and -nx < y hold for certain x e E, 0 < y e E and all n e]N, then n.sup (x,-x) Next we give some examples of partially ordered linear spaces. All examples which are given here are well known in the literature and there exists a reasonable uniformity in the literature about a standard name of most of them. 2.4. Examples. 2 (a)] R isth erea l plane partially ordered componentwise, i.e. a ttn e same (x,,x2)< (yi.y2)i f* i< Yi and* 2< y 2 time. Provided withth eusua l algebraic operations1 R isa Ries z space,wher e (z,,Zp)= (x ,,x 2)V(y,,y2)i fz .i sth emaximu mo fx .an dy .( i= 1,2) . Note thatK iseve n Dedekind complete. (b) (F, lex )i sth erea l plane ordered lexicographicly, i.e. (Xj.Xg)< (y1,y2)i f[ X j< yj ]o r[ x := yl &x 2< y 2]. Withth eusua l algebraic operations (F, lex )i sa Ries z space which 2 is totally ordered. However (K, lex )i sno tArchimedean , because n(0,l)< (1,0)fo ral l ne IN . (c)P(1R )i sth elinea r spaceo fal l real polynomialso nth erea l axis with pointwise linear operationsan dpointwis e partial ordering, i.e. x< y i nP(R )i fx(t )< y(t ) for all te R . P(]R)i sa partiall y ordered linear space,whic h ismoreove r Archime dean. However,P(1R )i sno ta Ries z space becausexV ydoe sno texis t forx an dy incomparable ,whic h obviously existi nP(1R) ,e.g .x an d y with x(t)= 1 an dy(t )= t fo r all teP areincomparable . (d)C(X ) is the linear space of all continuous real valued functions on a topological space Xwit h pointwise linear operations and pointwise partial ordering. C(X) isa n Archimedean Riesz space,bu t in general not Dedekind com plete. IfC(X )i s Dedekind complete then X is called extremally dis connected. (e) (cf.e.g . Luxemburg and Zaanen [1971,ex.ll.2(ix)]) . IfH isa Huber t space over the complex numbers,wit h inner product (.,.), then by Mw e denote the real linear space of all bounded Her- mitean operators on H,provide d with the partial ordering given by S< T for S,T e x if (Sx,x)< (Tx,x)fo r all xe H. 3Ci s an Archimedean partially ordered linear space.J Ci s a Riesz space only if the dimension of H is0 or 1. (fj (Compare e.g. Aliprantts and Burkinshaw[ 1978 ,ex .2.1 3 (2)]). If (X,r,y)i sa measur e space,i.e . a non-empty set Xan d aa-fiel d r of subsets of Xo n which isdefine d a non-negative countably additive measure u,the n letM(X,y ]b e the linear space of all real measurable functions on X. Ifw e provide M(X,u)wit h the partial ordering given by x< y if x(t)< y(t)fo r all te X,the n M(X,u,<) isa Dedekind a-complete Riesz space. (g) (cf.e.g . Luxemburg and Zaanen [1971,ex . 11.2(v)]). Ifi n M(X,u)fro m example (f)x is called equivalent with y (x^ y ) if x =y p-almost everywhere,the n^ isa n equivalence relation on M(X,u). The linear spaceM(X,y )o f all equivalence classes [x] in M(X,y) (natural algebraic operations) can be provided with a partial ordering given by[x ]< [y] ifx < yp-almos t everywhere.M(X,u,< ) isa Dedekind complete Riesz space. (h)s is the linear space of all sequences of real numbers.Wit h pointwise partial ordering s is a Dedekind complete Rieszspace . (i)b is the linear space ofal l bounded sequences of real numbers.Wit h pointwise partial ordering b isa Dedekind complete Riesz space. (j)c i sth elinea r spaceo fal l sequenceso frea l numbers which converge. With pointwise partial orderingc i sa nArchimedea n Riesz space,whic h isno tDedekin d complete,becaus ei fA = {(1,0,0,0,...),(1,0,1,0,0,...) , (1,0,1,0,1,0,0,...), ...}, thenA < (1,1,1,...). howeversu pA doe sno t exist. (k)c „i sth elinea r spaceo fal l sequenceso frea l numbers which converge to0 . With pointwise partial ordering cQi sa Dedekin d complete Riesz space. (1) cQQi sth elinea r spaceo fal l sequenceso frea l numbers whichar e eventually 0. With pointwise partial ordering cnn isa Dedekin d complete Riesz space. (m)F Ri sth elinea r spaceo fal l sequenceso frea l numbers which havea finite range.Wit h pointwise partial orderingF Ri sa nArchimedea n Riesz space whichi sno tDedekin d complete,becaus ei fA = {(1,0,0,0,...) , (1,^,0,0,0,...),(1,^,0,0,..),...} thenA < (1,1,1,...), however supA does notexist . 3. Elementary propertieso fRies z spaces In this section some abbreviationsar egiven ,mos to fwhic har ecommonl y used,furthe r some elementary propertiesar ederived . Fora nelemen tx o fa Ries z spaceE th epositiv e partx ofx i sdefine d byx =xVO ,th enegativ e partx "o fx b yx ~= (-x)V Oan dth eabsolut e value |x|ofx b y|x |= xV(-x) . xi ssai dt ob eorthogona l toy ,o rdisjoin tt oy ,i nformul axiy ,i f |x|A|y|= 0 .Th eorthogona l complement x ofa subse tX o fE i sdefine db y A ={ ye E;yi x foral l xG X};{x } isabbreviate dt ox . A subse tX o f Ean da subse tY o fE ar esai dt ob eorthogonal ,o rdisjoint ,i nformul a XIYi fxi yfo r allx e X an dy e Y. {x}lYi sabbreviate d toxlY . 3.1.Definition . A subset P of a Riesz space E is called a polar of E if P=P U. Iti s known that for every subsetX o fa Ries z spaceE th e equality X1 =X 111 holds (cf.e.g . Luxemburg and Zaanen [1971, thm 19.2(ii)]). This implies that every subseto f the form A (for some Xc E)i sa pola r of E;i ti seviden t that conversely every polar iso f this form. Every polaro fE i sa linea r subspace ofE (cf .e.g . Luxemburg and Zaanen [1971, thm 14.2]). For subsetsX an dY o fa Ries z spaceE w e useX ={ x;x e X} , X"= {x~ ;x e X} |X|= {|x| ; xe X},X+ Y = { x+ y ;x e X,y e Y} , X- Y = {x- y ; xe X,y € Y},XV Y= {xVy ;x e X,y e Y}an d XAY= {xAy ;x 6 X,y e Y}.W e abbreviate {x}+ Yt ox+ Y ,simila rabbre viations aremad ei nth e other cases. ForA e R we denote {Ax;x e X}b y AX. In every Riesz spaceE th eequalit yE =E -E holds (cf.e.g . Schaefer [1974,p .58]) .E + is called the positive coneo fE ;th e elementso fE ar e called the positive elementso fE . 3.2.Proposition . Forx and y positive elements of a Riesz space E the vncluszon x +y c (x+ y ) holds. Proof: Ifx +y i sfo r somes e Ethe n from0 < x ,0< y i tfollow s that xl s an dy 1 s .Bu t then alsou Is an dv i sfo r allu e x andv e y , henceu +v i. s ,whic h impliesu +v e (x+ y ) 3.3. Theorem,(cf .e.g . Aliprantis and Burkinshaw[ 1978 , thm. 1.1],Luxem burg and Zaanen[1971 ,cor . 12.3]an d Schaefer [1974,11, prop.1.4 ,cor .1 , cor.2 ] ) . Forx,y, Z elements of a Riesz space E we have (a)x = x +- x" , |x|= x ++ x~ ,x +A x ~= 0 (b)xv y= -((-x)A(-y)) ,xA y= -((-x)v(-y) ) (a) x+ (yvz )= ( x+ y)V( x+ z) ,x + (yAz )= ( x+ y)A( x+ z ) (d)A(xvy )= (Ax)V(Ay ) for all Ae IR ;+ |Ax |= \\\\x\forall AGIR (e)(xvy)V z= xv(yvz) , (xAy)Az= xA(yAz ) (f) x + y = xvy + xAy, |x - y| = xvy - xAy (9) (x - y)+ = x and (x - y)~ = y if xAy = 0 (h) xiy if and only if |x +-y | = |x - y| (i) (Birkhoff's identity) |xvz - yvz| + |xAz - yAz| = |x - y| (J) (x + y)Az < (xAz) + (yAz) if x,y,z e E+ (k) x< y is equivalent to x Thefollowin g theorem isfrequentl y used inRies zspac etheory . 3.4.Theorem , (comparee.g .Luxembur g andZaane n [1971,cor .15.6] ) (a) (Dominated décomposition property) If x, ,... ,x and y are positive elements of a Riesz space E such that y< x ,+ ...+ x holds, then there exist y,,..., y in E such that y =y ,+ ...+ y and y- 3.5.Theorem , (cf.e.g .Schaefe r[ 1974 ,I Ith m 1.5]) If X is a non-empty subset of a Riesz space E such that supX exists in E, then for every xe E also sup(xAX ) exists in E and the equality sup (xAX)= x Asu pX holds. Noww ecollec tsom enotion so fextrem eimportanc e inRies zspac etheory . 3.6. Definition. If E is a Riesz space, then (a) a linear subspace R of E is called a Riesz subspace of E if for x,ye R holds that xAye R . (b) a linear subspace J of E is called an (order) ideal of E if |x|< |y| for xe E and yG J implies xe J . (a) an ideal B of E is called a band of E if the following holds: if X is a subset of B such that supX exists, then supX e B . (d) an ideal J of E is called a o-band if for every countable subset X of J for which supX exists holds that supX e J . (e) an ideal J of E is called a principal ideal if there exists a ze E such that J= { xe E; |x|< Azfo rsom eX e R} . (f) a band B of E is called a principal band if there exists a ze E SUCT Î t/zatB = z U. (g) a band B of E is called a projection band if B+ B =E . (h) a band B of E which is a principal band and a projection band is called a principal projection band. Fora nelemen tz o fa Ries z spaceE th eidea l {xe E;|x|< Azfo rsom e Ae K } iscalle d theprincipa l ideal generated byz ,an di sdenote db yI. The bandz i1s1calle dth eprincipa l band generatedb yz ,an di sdenote d by Bz. Ifz i sa nelemen to fa Ries z spaceE ,the nth ese tIS(z )o fal l infinitely smalls with respectt oz (cf .sect .2 ) isa nidea l ofE ,becaus ei f x,y e IS(z)an dA, ye ] Rthe nfo ral l ne I Nw ehav e n|x|< zan dn|y |< z hence,fo r all ne]N,n|A||x| Note thatfo ra Ries z subspaceR o fa Ries z spaceE als o xVye R wheneve r xan dy ar ei nR .Hence ,an yRies z subspaceR o fa Ries z space E,wit hth e linear space structurean dth eorde r structure inherited from E,i sa Ries z spaceb yitself . Note further that every ideal isa Ries z subspace. Itfollow s from Luxemburgan dZaane n [1971, thm 19.2(i)] that every polar ofa Ries z spaceE i sa band; th econvers e implication holds underth e additional assumption thatE i sArchimedea n (cf.Luxembur g andZaane n [1971, thm 22.3]) . Asa nimmediat e consequenceo fth e definitionsw ehav e (cf.Luxembur gan d Zaanen [1971 , thm17.4 ]) . 3.7. Theorem. Any arbitrary non-empty set-theoretic intersection of Riesz subspaces (or ideals, or bands, or polars) is a Riesz subspace (or an ideal, or a band, or a polar). ByL(E), respectivelyR(E) ,1(E) ,B(E) ,F(E )w edenot eth ese to fal l linear subspaces,respectivel y Riesz subspaces,ideals ,bands , polarso fE ,eac h partially orderedb yinclusion . Asa consequenc e ofth m3. 7al lthes e sets form lattices under theiror dering. For,th einfimu mf otw oelement s isth eintersectio n ofthes e ele ments,th esupremu mo ftw oelement s isth eintersectio n ofal l linear sub- spaces, respectively Riesz subspaces, ideals,bands ,polar so fE whic h contain thesetw o elements. Iti sals oa nimmediat e consequenceo fth m3. 7tha tal lthes e latticesar e complete,moreove r 1(E),B(E )an dP(E )ar edistributiv e (cf.Schaefe r[ 1974 , II prop. 2.31 for1(E) , Luxemburg andZaane n [1971,th m22.6 ] forB(E),an d e.g. Bernau [1965a ,th m1 ] forf(E)) ;1(E ) andR(E)ar eno tdistributiv ei n general. We givea simpl e example for11(E) : TakeE = I R, R = {(x.,x Je E;x ,= x„} , S= {(xjj^ e E;X jeR] , T= {(0,x 2)e E;x 2e R } then (RVS)A(RVT)= E , but RV(SAT)= R,i f supremuman dinfimu m inA(E )ar edenote d byV an dA respectively.P(E)i sa complet e Boolean algebra (cf.Si k[195 6 ]o rBerna u [1956a ]);I(E)an dB(E )i ngenera l are not. 3.8.Definition . An ideal J of an Archimedean Riesz space E is called order dense if for all fs E with 0< f there exists ag e J with 0< g < f . In some representation theorieso fRies z spacesth efollowin g twonotion s playa nimportan t role. (cf.[Schaefer , IIIdef . 2.1,I Idef . 3.2]). 3.9.Definition . An ideal J of a Riesz space E is called a prime ideal if xe E,ye E and xAye J imply xe J ory e J . 3.10.Definition . An ideal M of a Riesz space E is called a maximal ideal if M=£ E and there is no ideal in E properly between M and E (i.e. any ideal J such that Mc J C E holds, satisfies either J= M or J= Ej . 3.11. Definition, (cf.A.L . Peressini [1967,cha p II,prop . 5.13b] ) A Riesz space E is called countably bounded if E contains a countable subset C with the property that for each xe E there exist eG C and Ae ] R such that x< Ae. E is called bounded if there exists ane e E such that for each xe E there exists aA € ] R such that x< Ae. In the last case e is called a strong order unit of E. 10 Iti seviden t that every bounded Riesz spacei scountabl y bounded;th e converse doesno thold ,fo rth eRies z space cQ0o fal l sequenceso frea l numbers with onlya finit e numbero fcomponent s not equal to0 , iscount - ably bounded (letC b eth esubse to fc Q0 whose elementsar e(1,0,0,0,..) , (2,2,0,0,..), (3,3,3,0,...) ),bu t cQ0i sno t bounded. Ina bounde d Riesz spaceE th eorde r unite ha s the property thate ={0} , fori fxle , then,i f|x |< Xe,w ehav e |x|AXe= 0 ,whic h impliesx = 0 . 3.12.Definition . A Riesz space E is called weakly bounded if there exists an e£ E with the property that e ={0 }. In that oase e is called a weak order unit of E. Iti seviden t that every bounded Riesz spacei sweakl y bounded andever y strong order uniti sa wea k order unit.Th econvers e does not hold becauseth e Riesz spaceE o fal l sequenceso frea l numbers hasa wea k order unit e= (1,1,1,...),bu tn ostron n order unit. The foregoing two examples canserv et odemonstrat e thata Ries z spaceE canb ecountabl y bounded without being weakly bounded, and weakly bounded without being countably bounded. The following theorem givesa nimportan t characterizationo fbounde d Archimedean Riesz spaces. 3.13. Theorem (cf. e.g. Luxemburgan dZaane n [1971,th m27.6]) . The intersection of all maximal ideals of a bounded Archimedean Riesz space consists of the zero element only. 3.14.Definition . A Riesz space E is said to have the projection property (abbreviated to PPj if every band of E is a projection band. E is said to have the principal property (abbreviated to PPPJ if every principal band of £ is a (principal) projection band. Noww eca nstat ea nimportan t theorem for Riesz spaces. 11 3.15. Theorem (cf.Luxembur gan dZaane n [1971, thm 25.1]) With obvious notational abbreviations the following implications hold in any Riesz space E. <* Ded. a-complete^ Ded. compl.^ PPP =» Arch. PP ** No implication in the converse direction holds; further E can have PP without being Dedekind a-complete and conversely; Dedekind a-completeness and PP together imply Dedekind completeness. Finally, we discuss briefly the notion of lateral completeness. 3.16. Definition. A Riesz space E is called (conditionally) lateral complete if for every (bounded) set D in E of pairwise disjoint elements sup D exists in E. We remark that the notiono flatera l completeness was already definedi n Nakano [1950] forDedekin d complete Riesz spaces. A fundamental breakthrough was achieved byVeksle ran dGeile r [1972], who proved that every Archimedean conditionally lateral complete Riesz spaceha s PP. Futher contributions areb yAlipranti san dBurkinsha w [1977], Bernau [1966 ],[ 1975 ],[ 1976 ],Bleie r [1976 ],Conra d [1969 ], Fremli n [1972 ], Jakubi k [1975],[ 1978 ]an dWickstea d [1979 ]. Fora Ries z spaceE latera l completeness andbein g Archimedean areindepen - 2 dent properties,fo r(] R, lex )i slatera l complete butno t Archimedean, C[0,1] isArchimedea n butno tlatera l complete. Fora nArchimedea n Riesz spaceE latera l completeness andDedekin d complete nessar eindependen t properties,becaus eth eRies z spaceo fal l bounded sequenceso frea l numbers isDedekin d complete,bu t not lateral complete. The Riesz spaceE o fal l real functions on] Rwhic har erigh t locally constanti never y te]R, (i.e. xeEif for all te K there existsa n E> 0 suc h that xi s constanti n[ t, t+ e) )i sa nexampl eo fa latera l complete Riesz space,whic h isno tDedekin d complete.Alipranti san d Burkinshaw [1978,ex . 23.30]an d Wickstead [1979] give exampleso fsuc h 12 a Riesz space;especiall y the examplei nAlipranti s andBurkinsha w [1978] is rather complicated. 3.17. Proposition. Every Dedekind complete Riesz space E is conditionally lateral complete. Proof: Evident 3.18. Proposition. Every lateral complete Riesz space E contains weak order units. Proof: Letsb eth ese to fal lsubset sX o fE o fpairwis e disjoint elements. S * 3.19. Definition. A Riesz space which is Dedekind complete and at the same time lateral complete is called universally complete or inextensible. Universally complete Riesz spacesar ever y important inRies z space theory; every Archimedean Riesz space admitsa uniqu e universal completion (cf. e.g. Conrad [1971 ]) . 4. Linear operators In this section,E an dF ar e arbirary Riesz spaces.Th ezer o operator fromE t oF wil l bedenote d by0 .Th eidentit y operatoro nE wil lb e denotedb yI^ ,o rsimpl yb yI .Fo ra linea r operatorT fro mE t oF the nullspace N(T)i sdefine d byN(T )= { xe E;T x= 0} .A linea r operator T fromE t oF i scalle d positive,i nformul aT > 0,i fT(E +)c F +. By £(E,F)w edenot eth elinea r spaceo fal llinea r operators fromE t oF ,provide d with the partial orderingS < T i fan donl yi f T- S > 0 ,th esocalle d operator ordering. Inth ecas eE = F th espac e £(E,F)ca nb egive n moreovera nalgebr a structureb ycomposition . Intha t case £(E,F)i sa partiall y ordered algebra, i.e.a nalgebr a which isa tth esam e timea partiall y ordered linear space,suc h thatth e producto ftw o positive elements isposi - 13 five again.A linea r operatorT e £(E,F)i scalle da Jorda n operator ifT i sth edifferenc eo ftw opositiv e linear operators.Th eclas s £ (E,F)ofJorda noperator si sa linea r subspaceo f£(E,F) . £(E,F )i s a partially ordered linear space underth eoperato r orderingan di n the caseE = F a partiall y ordered algebra.A linea r operatorT fro mE to Fi scalle d order boundedi fth eimag eo fever y orderbounde d subset ofE unde rT i sa norde rbounde d subseto fF .Als oth eclas s£ (E,F )o f all orderbounde d linear operatorsi sa linea r subspaceo f£(E,F )an da partially ordered linear space underth eoperato r ordering andi n the caseE = F a partiall y ordered algebra undercomposition . 4.1.Lemma . A mapping t from E to Y which satisfies (&) t(x+ y )= t(x )+ t(y ) for all x,y e E+ (b) t(Ax)= At(x ) for all'x e E and X >0 admits a unique extension to a linear operator T from E to F. If moreover the range of t is con tained in F , then T is positive. Proof: LetT :E -* -F b edefine db yT x= t(x +)- t(x") , thenT i sa linea r operator,whic hi sa nextensio no ft .I fs i sanothe r linear operator which isa nextensio no ft ,the ns x= s x -s x =t( x) - t(x~ )= Tx ,henc eS = T . If t(E+)c F + thenT(E +)c F +,henc eT i spositive . 4.2.Theorem , (cf.e.g . Schaefer[1974 ,I Vprop .1. 2]) . For a linear operator T from a Riesz space E to a Riesz space F for the assertions (a) J exists in £(E,F) (b) T is a Jordan operator (c) T is order bounded holds that (a)= *(b) , (b)= >(c) . In the case F is Dedekind complete all assertions are equivalent. 4.3.Theorem . (Riesz-Kantorovic,cf .e.g . Schaefer[ 1974 , IVprop .1. 3 ]). If E and F are Riesz spaces and F is Dedekind complete, then £fE,F ) is a Dedekind complete Riesz space, in which supT /orTc £ (E,F) such that T< is given by (supT)(x )= su p{T ^ +.. .+ T^ ; {Ty ..., T p}finit e subseto fT ,x, ,.. ., x >0-an dx = x ,+ .. .+ x n) (x >0 ) 14 Krengel [196 3] gives anexampl e of aJorda n operatorT from C[ -1 , 1] to C[-l,l ] forwhic h T does notexist .N oanswe r seems to have been given in the literature to the question whether PPfo r F isalread y sufficient to garantee that£ (E,F)i s aRies zspace . The following example shows that PP is notsufficient . 4.4. Example. IfE i s the Riesz subspace of sgenerate d bye = (1,1,1,...) and cQ0,the n E,wit h the Riesz space structure induced by s,i sa n Archimedean Riesz space,namel y the Riesz space of all eventually constant real sequences.Not e thate isa strong order unito fE . Iffo ral l me I Nth eelemen t e of Ei sdefine d by e (n)= & foral l m J nr ' m,n new (where Si sth e Kronecker function),the n the elements e,e,,e?)... form abasi s Bo f E.Le t a linear operatorT from Eto'F R be given by Te =0 ,Te ,= e ,an dT e =- e re ,fo r all n>2. 11 n n n n-1 n-1 For anelemen ty e FRw ewrit ey or (y) in stead ofy(n ) (ne IN) . Let 0< xe Eb earbitrary ,sa y CO x =A e+ 2 x.e.,the nA > 0 and A. >-X foral l ie IM ;almos t i=ln n 1 all A.ar e equal to0 . T isa Jorda n operator,becaus eT = I - (i-T), whereI isth e canonical embedding operator from Eint o FR,an d I-T is positive because 00 00 oo A. A . (I-T) x =A e+ 2 x.e. - s A,Te , =A e+ s (x.e.+ — e., - y 1e,) , i=2 nn i=2 1 n i=2 !n i-1 1_i n n so forn e I Iw e have ((I- T)x) n =A + \(^-) +X n+1(£) > A -A(^ ) -A(± )= 0 . SupposeT + exists in£ (E,FR), then for all neu we haveT e > (Te ) =- e , n n n n v n' n n' + hence (Tv e n') m> - n6 nmn, fom r all me IN . Iffo rcertai n pair (N,M)e I Nx I Nhold s thatH * M and (Te ) > 0,the n letS e £ J(E,FR)b edefine d by (Se) = (T+e) for all m * M, rn m + + + (se)M = (T e)M - (T eN)M, (sen)m = (T en)m for all (n,m)eux « such that (n,m)* (N,M)an d (SeN)M =0 . S > 0becaus e for all m * Mw e have (Sx) = (Tx) „ > 0an d x 'm v 'm 15 (sx)M = A(Se)M + ^ X^Se^ = A(Se)M + ^ A^se^ + XN(seN)M on 00 =X ( S> T becausefo ral lm *M w ehav e (Sx)m= ( Tx) m> (Tx)anm d (sx)M= x(se)M+ ^ xi(sei)M =A(se) M+ XM(seM)M+ .^ xi(sei)M i*M CO = MseM)M + X(S(e- e M))M + XM(seM)M + ^ Xi(Sei)M> X(SeM)M + XM(SeM)M > i^M > -y i +v = M M v 'M + S< T becausefo rm *M w ehav e (Sx)m= ( Tx) anm d oo ó> SX x Se + x Se x T+e x T+e + Se + Se ( )M = < >M . =l M i)M = ( >M - < A ^ M i)M V A i*N + X(T e)M- X^e^+^X^T+e.^ -U i^N + + s 16 (RX)K = X(Re)K + ^ X.(He.)K = X(Re)K + ^ X.(He.)K + XK(ReK)R + T+e + Re + =*(*> K jj V n>K V A =ïï r >è - I - °- i#K oo + R> T because (Rx)n = (T x)n > (Tx)n for n * K and (Rx)R-= X(Re)K + s ^(Re^ co X(ReK)K+ X(R(e- e R))K+ ^ X.(Re .) R+ XK(ReK)K>X(Re R)K+ XK(ReR)K> co + + R< T becaus e (Rx)n= (T x)n foral ln *K an d(Rx) R= X(Re) R+ _ 2 Xi(Rei)=K + + ^ r <(X+ XK)(T eK)K< co + + + X(T eK)K+ X(T (e- e K))K+ XK(T eK)K+ ^ X^\)K = co co + + + + + X(r\)K+ X(T (e- e K))K+ £ X.(? e.)K - X(T e)K+ ^ x.(T e.)K= (T x)K + + R< ï becausei = (ReK)K< (T eK)r + Itfollow stha t (T ev J nn=^ fo n ral l neu. + + Iffo rcertai npeMw e have (T e)p> (T ep)p, thenlet we £ (E,FR )b e such + + that (We)n= (T e)n foral ln * p ,(we ) =\ andWe p =T en foral l new. CO W> 0 becaus efo rn i=p w ehav e (Wx) =X(we ) + s A.(We. ) = CO CO + + (T x)n > 0 and (wx)p = X(we)p + _ï X.fwe^ = £ + s X.(T e.)p = £ +MTVp=F +r >0' 17 W >T because (Wx) =(T +x) > (T-x) foral l n *p an d F+i P (Wx)p =x(We) p + 2 A^We.^ =| + / > p =(Tx ) + + w oo oo + + + + =A + 2 \(T e.)F < A(T e)p + 2 X.(T e.)p = (T x)p. + + w 18 Chapter II SOME TYPES OF CONVERGENCE In this chapter three types of convergence are given;som e attention is paid to the relations between them and finally continuity of linear operators with respect to these types of convergence is defined. 5. Sequences in Riesz spaces In this section E is an arbitrary Rieszspace . A sequence in E is amappin g ffro mI Nt o E.A sequence f in E is called increasing,i n formula tf, iff(n )< f(n+l)hold s for all nG IN, an d decreasing, in formula 4if,i f f(n)> f(n + 1)hold s for all ne H . Wewrit e ftx ift f and sup f(N) = x,fl- yi f if and inf f(!N) =y . The class of all sequences in E is denoted by seq(E). On seq(E)w e define a linear structure by (f+g)(n)= f(n)+ g(n),(Af)(n )= Af(n )fo r all ne IN , if Xe F and f,gG seq(E). A partial ordering on seq(E) is defined by f< g ifan d only iff(n )< g(n) for all ne IN . (seq(E),< )i s a Riesz space inwhic h (fvg)(n)= f(n)Vg(n) for all ne ]N . In the sequel (seq(E),< )i s abbreviated to seq(E). For xe Ean d fG seq(E)w e define x+ fG seq(E)b y (x+f)(n)= x+ f(n) for all ne IN. The sequence in Ewit h range {0} isdenote d by 0. For fG seq(E)an d nG ]Nw ewrit e occasionally f or (f) instead off(n) . Iff is a sequence in Ean d a:I N^ I Ni sa strictly increasing function, then the sequence f°c is called a subsequence of f. Byn(E )w e denote the power set of E,i.e . the set {X;X c E} . 6. Order convergence The first type of convergence we discuss is order convergence. In this section Ei s an arbitrary Rieszspace . 19 We definea mappin g °L,calle d order limit,fro m seq(E)t on(E )whic h assignst ofe seq(E)th eelemen t ofn(E )consistin g ofal lx e E suc h that thereexist s age seq(E)wit h the property that |f- x |< g an d g;o. Inth ecas e °Lf i= 6.1. Theorem (compareLuxembur g and Zaanen [1971,th m 16.1 ]). For f,ge seq(E) andx, ye E,A, ye R it holds that (a) if ftx or f4-x then Lf= x (b) if t'for 4- fand Lf=x then ftx orf4- x respectively (c) if °Lf= x and °L g= y , then °L(Af+ yg )= A x+ y y (d) if °Lf= x and °L g=y then °L(fVg)= xV y and°L(fAg )= xA y (e) if f' is a subsequence off and Lf=x then Lf'= x (f) if 0< f < g and °L g= 0 then °Lf= 0 From this theorem itfollow s that theclas so f all order convergent sequences isa Ries z subspaceo f seq(E),and theclas so f all sequences order convergent to0 isa nidea lo f seq(E). 6.2. Theorem (compare Luxemburg and Zaanen [1971,sxc . 16.10] ) For a Riesz spaceE the following assertions are equivalent (a)E is Archimedean (b) if forxe E ,Ae R ,f e seq(E) anda e seq(R) holds that °Lf= x and °La= A , then °L(af)= Ax. Proof: (a) -*• (b) : Thereexist sa sequenc eg i nE suc h that |f- x [< g and g4-0. Ifa Q= su p |a(W)|,the nw e have foral ln e 1* 4 that 0< |a(n)f(n)- Ax |< |a(n)f(n )- a(n)x |+ |a(n) x-Ax| . Further ithold s that |af- ax |< a Qgan d acg4-0,henc eb yth m 6.1 (f)i t follows that°L(|a f- ax| )= 0 . Weals o have that L(|ax- Ax| )= 0 . Byth m 6.1 (c)i tfollow s now that°L(|a f- ax |+ |a x- Ax| )= 0 .On emor e applicationo fth m6. 1 (f) gives °L(|af- Ax| )= 0 ,henc e °L(af)= Ax . (b) -* (a): Suppose nx 20 anda e seqQR)ar e such that f(n)= x ,g(n )= y an d a(n)=— fo ral l neu, then °La= 0 ,henc e °L(ag)= O y= 0 . From0 < f< a g itfollow s nowb yth m 6.1(f) that Lf= 0 ,henc ex = 0 . 7. Regulator convergence Inthi s sectionE i sa narbitrar y Riesz space.W edefin ea mappin g L, called regulator limit,fro m the Cartesian product seq(E)x E to11(E ) which assignst o(f,u )e seq(E)x E theelemen t of11(E )consistin go f allx e E suc h that foral le> 0ther e existsa nN eI Nsuc h that |f(n)- x |< eu holds foral ln > N. L(f,u)i scalle d the regulator limito ff wit h respectt o regulatoru . Foru{ rL(f,u);u e E +}w ewrit e rLf. Iffo r somepai r (f,u)e seq(E)x E there isan xe E suc h that Y* Y* {x}= L(f,u), thenw ewrit ex = L(f,u). fe seq(E)i scalle d regulator convergenti f Lf * $. Itfollow s directly from thedefinition s that foral lu e E holds that L(0,u)= IS(u) . Ifu i sa stron g order unito fE ,the n L(0,u)= IS(E) , because ifn|x |< y fo rsom ex e E,y e E and alln e j| ,an dm e I Ni s such thaty < mu,the n n|x|< m u forai ln G IN,henc e n|x|< u fo rai l ne|, sox e IS(u). Hence,IS(u )= IS(E) . 7.1.Theore m (compare Luxemburg and Zaanen [1971,th m 16.2 (ii)] ) If x and y are elements of a Riesz space E and if f, ge seq(E) such that x€ Lf andy £ Lg., then (a) Ax+ u ye rL(Af+ yg ) for all A,yG M. (b) xVye rL(fVg) and xAy€ r L(fAg) r r (a) if f' is a subsequence of f and Lf= x then Lf'= x (d) if 0< f < g and 0€ r Lg then 0G rLf. From this theorem itfollow s that the class ofal l regulator convergent sequences isa Ries z subspaceo f seq(E)an d theclas s ofal l sequences which are regulator convergent to0 ,i sa nidea lo f seq(E). + r 7.2. Proposition,! ƒxe E ,ue E andf e seq(E) such that xe L(f,u) then rL(f,u)= x+ IS(u). Proof: ifz e IS(u), thenb y definition n|z|< u fo ral l neu. 21 xG rL(f,u),henc efo ral le > 0 ther eexist san w GI Nsuc h that |f(n)- x |< EUhold sfo ral ln > N . I fe > 0 the nfo ral ln e I Nsuc h 1 thatn > ma x (N1 ,entie r(•*-) )w e havetha t |f(n)- ( x+ z)|< |f(n)- x |+ |z|< Jeu+ le u= eu ,henc ex + z e rL(f,u). Conversely,i fy G L(f,u), thenfo ral le > 0 ther eexist san M GI Nsuc h that |f(n)- y |< e uhold sfo ral ln > M . Let e >0 .Fo ral ln > ma x (Nj, M t )w ehav etha t |x- y |< |f(n)- x |+ |f(n)- y |< Jeu+ Je u=eu ,henc ex - y G L(0,U)= IS(u). 7.3. Example. IfE = ( F, lex) ,u= (0,1 )an dfG seq(E)i ssuc htha t f(n)= (0»i )fo ral ln e IN ,the n rL(f,u)= {0 }becaus e0 G rL(f,u) and IS(u)= {0} .Howeve r IS(E)= {(0,A) ; XG]R}. Note that rL(f,(l,l))= IS(E )becaus e (1,1)i sa stron g orderuni to fE . 7.4. Proposition. If xG E andf G seq(E) such that x6 rLf,ifos n rLf= x + IS(E). Proof:i fz e IS(E), then there existsa y e E suc h that n|z|< y hold s foral ln e IN .x e rLfimplie s thatther eexist sa u e E such thatfo r alle > 0 ther eexist san N e] Nsuc h that |f(n)- x | 7.6.Theore m (cf.e.g . Luxemburgan dZaane n[1971 ,th m16. 2 (i) ]). V» In an Archimedean Riesz space E it follows from X= Lf for xG Land f G seq(E ) that x= °Lf . + r IfE i sa nArchimedea n Riesz spacean df G seq(E),u ,v G E, the n L(f,u) canb edifferen t from rL(f,v)i fu * v.A simpl eexampl et odemonstrat e 22 thisi sE =F , f(n)= ^ fo r alln eU , u= 1 ,v = 0 ,the n rL(f,u) =0 , r n however L(f,v)= 0 . Note thati nan yRies z spaceE w ehav e L(f,0) +0 fo rfe seq(E)i f and onlyi ff i seventuall y constanti n E. 7.7.Definitio n (cf.e.g . Luxemburgan dZaane n[ 1971 ,def . 39.1, def. 39.3an d def.42. 1] ). If uG E and fe seq(E) such that for all e> 0 there exists anN ef j suah that for all n,m >N holds that |f(m)- f(n) | < eu, t/zewf -i s called a U-Cauchy sequence in E. f£ se.q(E)i s called a regulator Cauchy sequence in £ if f is a u-Cauchy sequence in E for some ue E. E is called u-completeif for every u-Cauchy sequence f in £ holds that L(f,u) =£0. E is called regulator complete if E is U-complete for every ueE+. 7.8.Theore m (cf.Luxembur g andZaane n [1971,p .28 1(ii)]) . For every Riesz space E the following assertions are equivalent (a) E is Dedekind o-complete (b) E has PPP and E is regulator complete. 7.9. Theorem (cf.Luxembur gan dZaane n [1971,p .28 1(ii ) ]). For every Riesz space E the following assertions are equivalent (a) E is dedekind complete (b) E has PP and E is regulator complete. 7.10.Theore m (cf.Luxembur gan dZaane n [1971,th m 43.1] ) . The Riesz space C(X) is regulator complete for any topological space X. 8. Characteristic convergence The third notiono fconvergenc ew edefin e herei sth enotio no fcharacter istic convergence,whic h seemst ob enew .I nthi s sectionE i sa narbitrar y Rieszspace . 23 LetC L be themapping ,calle d characteristic limit,fro m seq(E)t on(E) , which assigns tof e seq(E)th eelemen to fn(E )consistin go f allx e E such that thereexist sa (countable )subse t{ P; n e H }o fth eBoolea n algebraJ>(E )o fal l polarso fEsuc h thatf(n )- x e p foral ln e IN, P 1c p foral ln e IN andn { p; n e n} = {o} ,f i scalle d character istic convergenti f cLf f 8.1.Lemma , If {P; n e IN} and {Q; n e M} are (countable) subsets of P(E) such that Pn+1c P^, Qn+1c Q ^ for all ne n andn {P p;n e IN} 1 L =n {P n;n e H}= n {Q n;n e IN} = {0} , thenn {(P n+ Q n) -; n e IN} = {0}. Proof:W eshal l denote suprema and infima inp(E )b yV an dA respectively . Noww e havefo r P,Q e P(E)tha t PAQ= Pn Q ,PV Q= ( Pu Q) 11 =( P+ Q)11 (cf. e.g.Berna u[196 5a ,proo fo fth m1 ] ). LetN e IN. I t follows from Bigard,Keime !an dWolfenstei n [1977,prop . ii 11 3.2.16] thatn {(P n+ Q n) ; n fle }=n {(P n +QJ ; n e IN, n > N } C n {(PN + Qn)U ; n e W } = A{PNVQn; n £ W } = V(A{Qn; n e W D ii = PjjV{0} = P It follows that n {(Pn + Qn) ; n e N}c n {p^; N e fl } = {0}, ii hencen {(Pn+ Q n) ; n e ]N} = {0}. 8.2.Theorem ,! ƒf e seq(E)-i s characteristic convergent then Lf= x /o r somex G E . Proof:Suppos e x,ye Lf,the n there exist (countable)subset s {Pp;n e H } an d {Qn;n e j\ }o ff(E )suc h thatf(n )- xe Pn> f(n)- ye Qn, P C P an d Q c fo ra1 1 n+1 n n+1 Qn ne IN andn {p^ ;n e H}= n{Q n;n e H} 11 = {0}.No ww e havex -y = f(n )-y - (f(n )- x )e Pn+ Q nc ( P +Q J for alln e n. Fro m the foregoing lemma itfollow s thatx = y . 8.3.Theorem . If x,y e Eand f ,g e seq(E) are such that x= Lf and y= c Lg, then (a) cL(Xf+ ug )= X x+ y y (b)c L(fVg)= xV yan dC L(fAg)= xA y (c) x= Lf' for every subsequence f'o /f . (d) cLg= 0 if 0< g< f an dC Lf= 0 . 24 Proof: Let {P; n e IN} b ea (countable )subse t ofP(E) such that f(n)- x e Pn,g(n )- ye Qn> Pn+1c P n and Qn+1c Q n foral l ne N, n {Pn; ne w} =n {Q n;n e IN} = {0}. (a) |Xf+ y g- (X x+ yy) |< |X(f- x )| + |y(g- y)| ,henc e for all ne H 11 u |Xf(n)+ yg(n )- (X x+ yy)|e Pp +Q nc (P n +Q^ . From (Pn+1+ Qn+1) iJ c c (Pn+ Q n) - foral ln e IN and lemma 8.1 itfollow s that L(Xf+ yg ) = Xx+ y y (b)Fro mBirkhoff' s identity (thru3. 3 (i)) and(a) itfollow s thatfo r alln e INw ehav e |(f(n)- g(n)) +- ( x- y) +| < |f(n)- g(n )- ( x-y )| G( Pn+ \^ 'nenc e °L^ ~9) += ( x" y) +- 0n emor e applicationo f (a) gives cL((f- g ) +g )= ( x- y ) +y , hence cL(fVg)= xVy . cL(fAg)= xA y follows similarly. (a) If a: H •* H isa strictl y increasing function,the nn { P.. ;n€ N} ={0} P C P fo ra1 1n e ' a(n+1) a(n) ^ and (f»a)(n)- xe Pa(n) foral l ne IN, henc e cL(f°a)= x . (d) Foral ln e INw ehav e0 < g(n )< f(n )e P , henc e cLg= 0 . Itfollow s from this theorem that the classo fal l characteristic con vergent sequencesi nE i sa Ries z subspaceo fseq(E) , moreover the class ofal l sequences which are characteristic convergent to0 i sa nidea lo f seq(E). 9. Comparison ofconvergence so fsequence s Inthi s sectionw e compare the foregoing three typeso f convergence. Inthi s sectionE i sa narbitrar y Rieszspace . The relation between order convergence and regulator convergence for sequences has been studiedi nLuxembur g and Zaanen[1971 ,§1 6] . Theirmai n results are thefollowing . 9.1.Definition . A Riesz apace E is called order convergence stable if for any fe seq(E) with flO there exists a 0< ae seq(R)3 such that ta, {a(n);ne W is not majorized and L(af)= 0 . 25 9.2.Theorem . An Archimedean Riesz space E is order convergence stable if and only if every order convergent sequence in E is also regulator convergent. 9.3.Theorem . Every regulator convergent sequence f in an Archimedean r o Riesz space E is also order convergent, moreover Lf = Lf. 9.4. Example. Theconditio n thatE i sArchimedea n cannotb eomitte d inth eforegoin g theorem,becaus ei nexampl e7. 3w ehav e L(f,u)= IS(E) for somef e seq(E)an du e E , whil e IS(E)i sno ta singleton . 9.5. Theorem. If for fe seq(E) andx, ye E holds that Lf =x and Lf=y then x=y . Proof:I ti ssufficien tt osho w thati fc Lf= 0 an d °Lf =x the nx = 0 . Thereexist s age seq(E)suc h that |f- x |< g an dgJ-0 ,henc e f(n)e [ x- g(n) ,x+ g(n )] fo ral ln efli . For all n e IN we have [x - g(n + 1), x + g(n + 1) ].c [x - g(n), x + g(n)] and n [x - g(n), x + g(n) ] = {x}. Onth eothe rhan d f(n)e p v ' wit hnP eD(E )nsuc •vh' tha tP ± 1 cP fon+ral 1 l n ne h and n{Pn;n G IN }= {0} . Itfollow sno wtha tx = 0 . 9.6. Example.No t every characteristic convergent sequencef i na Ries z spaceE i sautomaticall y order convergent.I fE = C[0, 1] an df e seq(E) issuc h that (f(n))(t)= -n 2t+ n i ft e [0,n_1] an d(f(n))(t )= 0i f _1 ii U te [n ,l ]fo ral l nel, then n{f(n) ;n e J O ={0} , f(n)e f(n) andf( n+ l)11 c f(n)11 foral ln e IN ,henc e cLf =0 . But Lf= 0 because (f(n);n e IM }i sno tbounded . Nextw esho w that regulator convergencean dcharacteristi c convergence dono timpl yeac hother . Ifi na narbitrar y Archimedean Riesz spaceE * {0}w ehav ex > 0 ,the n the sequencef i nE wit h f(n)= n xfo ral ln e I Ni sregulato rconver gentt o{0 }(regulato rx) . Suppose thatf i scharacteristi c convergent,the nf i scharacteristi c convergentt o{0 }b ythm s9. 3an d9.5 . However,f(n ) =( —x ) =x foral ln e IN ,henc ef i sno t characteris tic convergentt o{0} ,contradiction . 26 Conversely,i fE i sth e(Dedekin d complete)Ries z spaceo fal l sequences xo frea l numberswit h componentwise linearoperation san dcomponentwis e partial ordering,suc h that |x|i sbounde db ya rea l multipleo f r= (1,2,3,... )an df i sth esequenc ei nE suc h that f(n)= (1,2,3,..,n,0,0,0,.. )fo ral ln e]\| , then,i fg = f - r ,w ehav e f(n)- r e g(n)iJ-,g( n+ 1)U cg (n)U foral l n,£Kan d n{g(n)n; ne IN }= {0} ,henc ec Lf= r . Supposef i sregulato r convergent,the n Lf= r b ythm s9. 3an d9.5 . Ifu i sth ecorrespondin g regulator,sa yu< Ar,the n certainlyA ri s regulator,bu tthe n alsor i sregulator . Itfollow s that foral l e> 0 ther e existsa H e fi such thatfo ral l n >N £ ithold s that |f(n)- r |< er;however ,i fw etak e e=i, then theredoe sno t exista ne u such thatfo ral ln > N hold s that |f(n)- r\ < |r,becaus e thiswoul d imply that f(n) > |rfo rn >N , contradiction. Hence,f i sno tregulato rconvergent . From the foregoingi tfollow s thatorde rconvergenc e doesno timpl y characteristic convergence. 10. Some notionso fideal s Inth eseque lw enee d some carefully chosen notionso fideals ,whic har e defined below. In this sectionE i sa narbitrar y Rieszspace . Fora nelemen tx e E w ewrit eI ={ ye E ; |y|< A xfo rsom eA e R} (sect.3) . 10.1. Definition. A d—ideal of a Riesz spaoeE is an ideal J of E with the property that x =y andx eJ ;y e E imply yG J . The notiono fd-idea l seemst ohav e been introducedb yBal l [1975] unde r the name full convex Jl-subgroup. 11 10.2. Proposition, (cf.Bal l [1975,th m1. 1(ii) ]) An ideal J of a Riesz space E is a d-ideal if and only if x cJ for every xe J . 10.3. Proposition. Every band B of a Riesz space E is a d-ideal. Proof:i fx e B ,y e E an dx =y ,the nb yLuxembur g andZaane n [1971,th m24. 7 (ii)] ithold s that |y|= sup{|y |A n|x| ;n G M} .Fo ral l ne Uwe have |y|A n|x |e B ,henc e also |y|e B ,s oy e B . 10.4. Example.No tever y d-ideal ofa Ries z spacei sa band .Le tEb e the Archimedean Riesz spaceo fal lcontinuou s functionsx o na locall y compact Hausdorff spaceS ,whic h isno tcompact .I fJ i sth eidea l ofE consistin g of all xe E whic h havea compac t support,the nJ i sa d-idea l ofE , because ify e x forxe J ,the nth esuppor to fy i scontaine d in the supporto fx ,henc ei scompact .Bu tJ i sno ta ban d becauseJ i sno ta polar,fo rJ =E an dJ fE . 10.5. Definition. An ideal J of a Riesz space E is called (a) an o-ideal if for all 0< x G J holds that ye J whenever ye Lf for some 0< f£ seq(I ) such that +f. x r (b) a r-ideal if for all 0< x e J holds that ye J whenever ye Lf for some 0< f e seq(I ) such that tf. r (a) a c-ideal if for all 0<*e J holds that ye J whenever ye Lf for some 0< f £ seq(I ) such that tf. or c x ( L, L and L are taken in E). We note that therei sa clos e relation between r-idealsdefine d here, and z-ideals,whic har edefine d byHuijsman san dD ePagte r [1980] . 10.6. Proposition. Every d-ideal J of a Riesz space E is an o-ideal. Proof:i f0 < x e J ,0 < f e seq(I ),+ fan dy = °Lf ,the ni tfollow s fromth m6.1(b) thatf + y .Fo ral ln e H wehav e that f(n)e I cx 11. il 11 Fromx isa ban d itfollow sno wtha ty e x ,henc ey e J . Adetaile d discussion ofth emutua l connections between o-ideals, r-ideals andc-ideal si splanne d forth enea r future. 28 11. Continuityo flinea r operators In this sectionw estud y continuityo flinea r operators with respectt o the three typeso fconvergenc e defined above.E an dFar earbitrar y Riesz spacesi nthi s section. Every linearoperato rT fro mE t oF induce s a linear operator,als o denotedb yT , form seq(E)t oseq(F )b y(Tf)(n ) =T(f(n) )fo ral ln e n . 11.1. Definition. A linear operator T from E to F is called (a) (sequentially) order continuous (or an integral operator) if for every fG seq(E) holds that °L(Tf.)= 0 whenever °Lf =0 . (b) (sequentially) regulator continuous if for every fe seq(E) holds that rL(Tf)= IS(F )whenever rLf =IS(E) . (a) (sequentially) characteristic continuous if for every fG seq(E ) holds that CL(Tf)= 0 whenever cLf = 0. 11.2. Example.No tevery positive linear operatorT fro ma Ries z space Et oa Ries z spaceF i sorde r continuous,becuas ei fT i sth elinea r operator fromC [0, 1] t oR which assignst ox th evalu e x(0),the nT ispositive ,however ,i ff e seq(C [0,1 ] ) i ssuc h thatfo ral l ne W we have (f(n))(t)= 0 i ft e [n _1,l] an d(f(n))(t )= 1 - n t if te [0,n_1], the nf 4 -0 , hence °Lf =0 .However ,°L(Tf )= 1 becaus e (Tf)(n)= 1 fo ral ln e ]N. 11.3. Theorem, (compare Vulikh [1967, thmVII I1. 2]) . Every Jordan operator T from a Riesz space E to a Riesz space F is regulator con tinuous . Proof: Iti ssufficien tt o givea proo ffo rpositiv eT only . If rLf =IS(E )the n0 e rLf,henc e there existsa u G E+ such thatfo r all e> 0 ther e existsa N e IN such thatfo ral l n> w holds that e e |f(n)|< eu.Bu tthe n also |Tf(n)|< T|f(n)|< eT u holdsfo ral ln > N , hence0 e rL(Tf,Tu)c ri_(Tf).B yprop .7. 4w ehav eno wtha t rL(Tf)= IS(F). 11.4. Example.No tever y positive linearoperato rT fro ma Ries z spaceE toa Ries z spaceF i scharacteristi c continuous,becaus ei fT i s the canonical embedding operator fromC [0, 1] int oth eRies z spaceF o f all real functionso n[0, 1 ],the nT i sa positiv e linear operator. 29 Iff e seq(E)i ssuc h thatfo ral ln e N andt e [0, 1] w ehav e (f(n))(t)= 0 i ft e [n_1,l] an d(f(n))(t )= -n t+1 i fte [0,n _1 ], then cLf =0 ,becaus e f(n)e f(n)U foral l ne IN, f(n+l) ii c f(n)ii for alln e M andn {f(n)11;n e H } ={0} .Bu t(Tf(n)) 11 D XU for alln e j)wher ex e F i ssuc h that x(0)= 1 an dx(t )= 0 fo rte (0,1] , hence CI-(Tf) f0 . Itma yb equestione d whether every integral operatori sorde r bounded. Fora rathe r extensive classo fintegra l operators (not exhaustive) this questioni sanswere d byPeressin i [1967,prop .I .5.15 ,prop .I .5.13 b ], wherei ti sprove d thatever y integral operator froma nArchimedea n Riesz spacet oa nArchimedea n countably bounded Riesz spacei sorde r bounded. 11.5. Example.No tever y positive linear operatorT fro ma Ries z space Et oa Ries z spaceF i sa nintegra l operator.Le tE= C [0, 1 ], F = F , T:E • +F suc h thatT x= x(0) ,the nT i sa positiv e linear operator.I f fe seq(E)i ssuc h thatfo ral ln e M wehav e (f(n))(t)= 0i f te [n _1,l] an d(f(n))(t )= 1 - n ti ft e [0,n _1 ], the n °Lf= 0 , but Tf(n)= 1 fo ral l ne n, hence °i-(Tf)=1 . 11.6. Theorem. If E and F are Archimedean Riesz spaces and E is order convergence stable, then every order bounded linear operator T from E to F is an integral operator. Proof:I ffo rfe seq(E)hold s that °Lf= 0 ,the n there existsa ge seq(E)suc h that |f|< g an dg + 0 .E i sorde r convergence stable, hence there existsa 0 < a e seq(R) such that ta, (a(n);n e K} i s notmajorize dan d°i.(ag )= 0 .Th elatte r impliesth eexistenc eo fx e E such that |ag|< x . Noww ehav e that i'a(n)f(n) ;n e M} isorde r bounded, because- x< -ag< a f< a g< x.Bu tthe n alsoT({a(n ) f(n);ne K} i s order bounded inF ,sa y- y< T(a(n) f(n))< y fo r alln e n . Itfollow s that |T(f(n))|< (a(n))-1yfo ral ln e H such that a(n) f 0.Sinc eF is Archimedean,w ehav e °i-(Tf)= 0 b yth m6. 2 (b). 11.7. Example.I fa nArchimedea n Riesz spaceE ha sth epropert y that every order bounded linear operatorT fro mE t oa narbitrar y Archimedean Riesz spaceF i sa nintegra l operator,the nE i sno tnecessar y order convergence stable,eve n ifE i suniversall y complete. 30 Todemonstrat e thisw e use aRies z spacewhic h appears inTucke r[ 197 4] . If Ei s the Riesz space (with pointwise linear operations and pointwise ordering)o f all realvalued functions on the set So f all 0< aG seq(R) such that fa,a(l )> 0an d {a(n); ne M} i s notmajorized ,the n Ei s universally complete. Itfollow s from Fremlin[1975 ,cor . 1.13 ] that every order bounded linearoperato r from E toa n arbitrary Archimedean Riesz space F isa n integral operator.However ,E isno t order conver gence stable,because ,i ff G seq(E)i s such that (f(n))(a)= (a(n)) , then f 4-0 becaus e inf{(f(n))(a); n G fi} = 0 fo r alla e S,a s (f(n))(a)= (a(n))"1 for all ne IN. But if0 < a,fa , a(l)> 0an d {a(n); ne K} is notmajorized , then for all ne IN we have that a(n)(f(n))(a)= 1,henc e L(af) f 0.Thi s implies that E is not order convergence stable. 31 Chapter III DISJUNCTIVE LINEAR OPERATORS In this chapterw e study disjunctive linear operators,especiall y ortho- morphisms and disjunctive linear functionals. Furtherw e give some examples ofunbounde d orthomorphisms. 12. Orthomorphisms. A notion of rather recent origin is the notion of disjunctive linear operator.Th e study of some special types ofdisjunctiv e linear operators such as Riesz homomorphisms,i smuc h older. 12.1. Definition, (cf.Cristesc u [1976,p . 186] ). A linear operator T from a Riesz spaceE to a Riesz spaceF is called a disjunctive linear operator if for all x,y e E holds that Tx1T ywhenever x1y . 12.2.Theorem . For a linear operatorT from a Riesz spaceE to a Riesz spaceF the following assertions are equivalent (a)T is a disjunctive linear operator (b) |T|x|[= |Tx | for allx Ee Proof: (a) •*(b): x 1x , s oT x 1T x. No w |TX|= |T X -T x| = |Tx++ Tx" | = |T|X||b yth m3. 3 (h). (b) -> (a): ifx ly the n |xA| |y |= 0 . Now |T|X|- T|y | = |T (| |x |- |y || = |T(|x|- |y|) ++ T(|x |- |y|)" |= |T|X |+ T|y|| ,becaus e (|x|- |y|)+ = |x|an d (|x|- |y|) "= |y |b yth m 3.3 (g).B yth m 3.3 (h)w e have Tjx|i T|y| , hence |T|X|| I JT|y||,s o|TX |l |Ty|,consequentl y Txl Ty. Themos t important disjunctive linear operators are the positive ones, called Riesz homomorphisms.A bijectiv e Riesz homomorphism is called a Riesz isomorphism;a Ries z spaceE i s called Riesz isomorphic toa RieszspaceF i f there existsa Ries z isomorphism fromE t oF . With theai d ofRies z homomorphisms factor spaceso f Riesz spaces can bedefined ,whic h are Riesz spaces themselves (cf.e.g . Luxemburg and Zaanen [1971,§1 8 ]). This isa consequenc eo f the preservationo f the lattice operations bya Ries z homomorphism,a fac twhic h is stated below. 32 12.3. Theorem. For a linear operator T from a Riesz space E to a Riesz apace F the following assertions are equivalent (a) T is a Riesz homomorphism (b)T(xVy )= TxVT y for allx ,y eE (c) T(xAy)= TxAT y for allx ,y 6E (d) |Tx|= T|x | for alxl eE . (compare Schaefer [1974, II,prop .2. 5 ]wher e similar statements are proved; Riesz homomorphism are called lattice homomorphisms there). There existsa nimportan t relation between realvalue d Riesz homomorphisms ona Ries z spaceE an dmaxima l ideals ofE .Thi s relationship is expressed in the following theorem. 12.4. Theorem, (compare e.g. Luxemburg and Zaanen[1971 ,th m 27.3 (i)] and Schaefer[ 1974 ,cor .o f II.prop .3. 4 ] ). If <$> is a realvalued Riesz homomorphism on a Riesz space E, then the nullspace N((f>)of § is a maxi mal ideal of E. If Mis a maximal ideal of E and x£ E is arbitrary such that x£ M , then there exists exactly one realvalued Riesz homomorphism <|> onE such that 12.5. Definition. A realvalued Riesz homomorphism § on a Riesz space E with strong order unit e is called standard if 12.6. Theorem. For an Archimedean Riesz space E with strong order unit e the set R is total, i.e. R is not empty and if <)>(x)= 0 for certain xe E and all 12.7. Example, (cf.Meye r[1979 ,Ex . 1.4 ]). LetE b e the Riesz subspace ofC [0, 1 ] consisting of allx e C [0, 1 ] such that the rightdifferen tial quotient ofx i nth e pointt =- ^ exist sa sa rea l number x'(-~). 33 letT :E • +] Rb eth elinea r operator such thatT x= x'( T). Ifxl y i nE , 11 thenx'^ )=0 o ry' W =0 ,henc eT xl Ty ,s oTi sadisjunctiv e linear operator. Note that neitherT no r- Tar eRies z homomorphisms. 12.8.Theorem ,(cf .Meye r[1979 ,Th mI .6 ]). A disjunctive linear oper atoris a Jordan operator if an only if it is orderbounded. 12.9.Definition .A linear operatorT from a Riesz spaceE to itself is called a stabilizer onE if T preserves orthogonality in the fol lowing strong sense: if x1y then alsoT x1y . 12.10.Theorem .For a linear operatorT from a Riesz spaceE to itself, the following assertions are equivalent (a) T is a stabilizer on E (b)T is polar preserving, i.e. T(P)cP for every polarP of E (c)T xe x iJ- for every xeE . Proof: (a)• + (b):i fx e P ,the nx l y fo ral ly e F, henc eT xl y fo r ally e r, orT xl P1,henc eT xe P11= P (b) -> (a): evident (a) ->- (a): Supposex l y .T xe x ,•henc eT xl y . The linear subspaceo f£(E,E )consistin go fal lstabilizer so nE i s denotedb yStab(E) . Insectio n4 i twa sobserve d that£(E,E )i sapar tially ordered algebrai fw esuppos eo n£(E,E )th e operator orderingan d ifmultiplicatio n isth e compositiono fmappings .Stab(E )i sa partiall y ordered subalgebrao f£(E,E )becaus ex ,ye E,x i y an dS ,Te Stab(E) imply Sxl y ,henc eTS xyi . Until recentlyi twa sunknow n whether every stabilizer isals oa Jorda n operator.A negativ e answert othi s questionwa sgive n independentlyb y Meyer[ 197 9] an dBerna u [197 9] .Thei r counterexamplesar eessentiall y the samean di nfac ta modificatio no fawel l known (norm)unbounde d linear operator,namel yth e differential operatori na nappropriat e Hubert space. Really surprising isth e fact that there exist also unbounded stabilizers in some universally complete Riesz spaces.Thi swa sprove db yWickstea d [1979] b ya kin do fHahn-Banac h prooffo rth eexistenc eo fa n(unbounded ) extensiono fth eMeyer-Berna u stabilizert oth euniversa l completiono f 34 the underlying Riesz space.Abramoviï ,Veksle ran dKolduno v [1979] assert that there existsa bijectiv e unbounded stabilizero nth eRies z space M([0,1 ],y ,< )wher ey i sLebesqu e measure (example 3.4 (g)) . We givemor e exampleso funbounde d stabilizers insectio n14 . The fact that there exist unbounded stabilizers implies that Stab(E) for a Riesz spaceE i si ngenera l nota Ries z space,becaus efo revery Riesz spaceE hold s thatE = E -E (every elemento fa Ries z spaceca n be writtena sth edifferenc eo ftw o positive elements,sectio n3) . 12.11.Definition . A linear operator on a Riesz space E which is the difference of two positive stabilizers on E is called an orthomorphism onE . Note that every positive orthomorphism isa Ries z homomorphism. The linear subspaceo f£(E,E ) consistingo fal l orthomorphismso nE is denotedb yOrth(E) . Orth(E)i sa partiall y ordered subalgebrao fStab(E) . In contrastt oStab(E )w ehav e thatOrth(E )i sa Ries z spacei ngeneral , wheneverE i sArchimedean . Thiswa sindependentl y proved byBigar dan d Keimel [1969] an dConra dan dDie m[197 1] .A direc t proofwa s givenb y Bernau [197 9] .Howeve rth elas t proofi srathe r complicatedan dno t very transparent. A linear operatorT fro ma Ries z spaceE t oitsel fi scalle da ûêntre operator onE i fT i sbounde d inth eoperato r ordering of£(E,E )b ytw o multipleso fth e identity operator l£o nE ,i.e .i fther e exist A,y e I R such that Al< T < yl. Every centre operatorT o na Ries z spaceE i sa northomorphism , because if AI< T < ylfo rA ,ye R thenT i sth edifferenc eo f|y| lan d |y|l- T .|y| li spositiv ean da stabilize r becausex l y implie s |y|x l y . Also |y|l- T i spositiv e becauseT < yT< |y|lan da stabilize r because ifx l y then |y||x|A|y |= 0 ,henc e (|y| l- T )|x|/\|y |= 0 ,s ocertainl y |(|y|l- T)x|A|y |= 0 ,henc e (|y|l- T) xl y . The linear subspaceo f£(E,E ) consisting ofal l centre operatorso nE is denoted byZ(E) .W enot e thatZ(E )i sa partiall y ordered subalgebra 35 of Orth(E). Itfollow s immediately fromth edefinition s that Z(E)i s a Riesz subspaceo fOrth(E) ,wheneve rE i sa nArchimedea n Rieszspace . 12.12. Example.I fE i sth epartiall y ordered linear spaceo fal lfunc tionsx fro ma narbitrar y non-emptyse tSt oE , with pointwise linear operationsan dpointwis e partial ordering,the nE i sa universall y com plete Riesz space.I fw edefin ea nalgebr a structureo nE b ypointwis e multiplication,the nE i sa partiall y ordered algebrawit h multiplicative unite ,th efunctio n constant1 o nS . Iffo rze E w edefin eth elinea roperato rfl b fly x= xz ,the nfl i s a stabilizer,becaus ei fx l y i nE ,the n x(s)y(s)= 0fo ral ls e S ,henc e x(s)z(s )y(s )= 0 an dthi s impliesx zi y ,o fir xi y .I nfact ,fl i s an orthomorphism,becaus efl fl= +- A anfld + an dfl ar epositiv e linear operators. Every stabilizerca nb ewritte n converselya sa noperato rfl fo rsom e ze E,henc e isa northomorphism . Thisca nb esee na sfollows .I fTi s a stabilizero nE an dz = Te ,the nfo rever yx e Ean dever ys e s w e have (x- x(s)e)(s )= 0 ,henc ex - x(s) el x where xAt) =0 i ft/ s , Xs(s)= 1 . Itfollow s that alsoT( x- x(s)e )l x >henc e (Tx- x(s)z)(s )= 0 ,whic h implies that (Tx)(s)= x(s)z(s) . HenceT fl= . Iti son eo fou rpurpose st ofin da descriptio no forthomorphism sa s multiplication operators ina narbitrar y Archimedean Riesz space.Whil e notever y Archimedean Riesz spaceca nb eprovide d witha nappropriat e multiplication,w eshal l deal inth eseque l with multiplicationsi n Riesz spaces,whic har eonl y partial ina certai n sense.I norde rt o developa nindependen t theory,w eshal l maken ous eo fth efac t that Orth(E)i sa Ries z spacewheneve rE i sa nArchimedea n Rieszspace . 12.13. Proposition, (compare Conradan dDie m[1971 ,Prop .2. 1 ]) If T is a positive linear operator from a Riesz space E to itself, then (a) T is an orthomorphism whenever I+ T is a Riesz homomorphism (b) I+ T is an orthomorphism whenever T is an orthomorphism. Proof: (a) ifI + T i sa Ries z homomorphisman dx l y i nE the n (I+ T)|x|A( I+ T )|y | =0.Fro m |TX| 36 |y|< (I+ T) IyI i tfollow sno wtha t |Tx|A|y|= 0 ,henc eT xiy . ffcji fT i sa northomorphis man dx l y the nw ehav eT x1 y an dxl y , hencex + T x1 y ,o r( I+ T) xi y . 12.14. Proposition. If T is a positive orthomorphism on a Riesz space E then I+ T is an infective orthomorphism onE . Proof:B yth eforegoin g propositionI + T i sa northomorphis mo n E. Ifx ,y >0 i nE ,the n from (I+T) x= ( I+T) yi tfollow s that y -x = T( x- y) , hence (y- x) + = (T(X -y)) + =T( x- y) + e( x- y) + . But also (y- x ) =( x- y) " e (x- y) ~ .I tfollow s that (y- x ) e( x- y ) n (x- y) ~ ={0} ,henc ex > y .B ysymmetr y also y> x ,henc e x=y. Ifx ,y e E ar earbitrary ,the n from (I+ T) X =( I+ T) y itfollow s that( I+ T)x += (( I+ T)x) += (( I+ T)y) += ( I+ T)y +,henc eb y the foregoingx =y , an dsimilarl yx ~= y" ,s ox =y . Thefollowin gtheore mi sadirec tconsequenc eo fLuxembur gan dSche p[1978 ,th m1. 3] 12.15.Theorem . Every orthomorphism T on an Archimedean Riesz space E is order continuous. In this section 14example sar egive n which show thatth m12.1 5doe sno t holdfo rarbitrar y stabilizers. 12.16.Theorem . Every stabiliser T on a Riesz space E is characteristic continuous. Proof:i fc Lf= 0 fo rsom ef e seq(E),the n there existsa (countable) subset {P; ne }|) o fth eBoolea n algebraP(E )o fal l polarso fE suc h thatP n+1c P nfo ral ln e IN, f(n )e P nfo ral l ne Nan d <~i{P p;n e IN}= {0} .No wb yth m12.1 0w ehav e thatT P cp forever y neu, hence also(T P) UCP foral ln e n. n n ii M u Itfollow s thatTf(n )e (TPn) foral l ne W, (TP+1 ) C(T P) for U U all ne ] Nan dr> {(T P ) ; ne N} cn{p ; ne n }= {0} ,henc e n UTPn)11; ne ]N}= {0} ,s oc L(Tf)= 0 . 12.17.Theorem . Every orthomorphism T on an Archimedean Riesz space E preserves all c-ideals of E, i.e. TJC J for every c-ideal J of E. Proof:I ti ssufficien tt ogiv ea proo ffo rpositiv eT only . 37 LetO < x e J .Defin ef e seq(Ix)b yf(n )= TxAn xfo ral l ne IN,the n 0< f an d-t-f .Furthe rw ehav e |f(n)- Tx |= (T X- nx) + foral l ne f\. Ifw edefin e P =(T x- nx) + foral l ne M , then f(n)- T xe p for alln e n. Als oP n+1c P nfo ral ln e IN.n {pn;n e j|}= {0} ,becaus e iffo r0< v e Ehold s that0 < v e (TX- nx )+i i foral l ne IN, the n vi (Tx- nx) ~fo ral ln e f|, henc ev i (n~T x-x)~ , sov i (x- n "Tx )+ foral ln e K. EArchimedea n implies thatx = sup{( x —Tx)+;n e n} , henceb yth m3. 5i tfollow s thatv i x,s ov l TX.Bu tthe n certainly vi (Tx- nx ), henc ev e (TX -nx) + forai ln e n. i tfollow s that v= 0 .Hence ,c Lf= Tx .c Lfe J becauseJ i sa c-ideal ,thu sT xeJ ; we conclude thatT Jc J . 12.18.Theorem . Every orthomorphism T on an Archimedean Riesz space E preserves all o^ideals of E, i.e. TJ c J for every o-ideal J of E. Proof:I ti ssufficien tt ogiv ea proo ffo rpositiv eT only .Le t0< x e J . Definef e seq( I )b yf(n )= TxAn xfo ral ln e u ,the n0 < f an dtf . 11 Byth m12.1 0w ehav eT xe x ,henc eT x= sup{f(n) ;n e n} .Fro m thm6. 1 (a)i tfollow sno wtha tT x= Lf, henceT xe J ,s oT Jc J . 12.19. Remark.No tever y orthomorphismT o na Ries z spaceE preserve sal l idealsJ o fE .I fE i sth eRies zspac es o fal l sequenceso frea l numbers andJ i sth eidea lo fal l sequences convergingt o0 ,e = (1 ,1 ,1 ,... )eE , y= (1 ,-£ ,-j , ...)e J an dTe Orth(E )i ssuc h that (Tx)(n)= nx(n )fo ral l ne M ,the nT y= e £ J ,henc eT Jf. J . 12.20. Theorem. (Bigard,Keime ian dWolfenstei n [1977,th m12.2. 7] ) If T is an orthomorphism on an Archimedean Riesz space E, then the nullspace N(T) of T is equal to T(E)- . We note thatth eproo fi sb yelementar y means. Adirec t consequenceo fthi s theoremi s 12.21. Theorem, (compareBigar d[ 197 2] ) If S andT are orthomorphisms on an Archimedean Riesz space E which coincide on a subset X for which X =X. , then S= T . Proof:X c N( S- T) , henceE = X U cN( S-T) U= N(S-T), soS = T 38 Orthomorphismso nArchimedea n Riesz spaces with strong order unit have special properties. Herew eshal l prove twoo fthem . 12.22. Theorem. If E is an Archimedean Riesz space with strong order unit e, then TMc M for every Te Orth(E) and every maximal ideal M of E. Proof: Iti sn orestrictio n togiv ea proo f for positiveT only. Ri s the seto fal l standard realvalue d Riesz homomorphismso nE . LetM b e an arbitrary maximal ideal ofE .B yth m12. 4 there existsa Inthi s sectionw edetermin eth enullspace so fdisjunctiv e linear functionalso na non-trivia l Riesz spaceE . Itappear s thatth eform so fthes e nullspaces resembleth enotion s of prime idealsan dmaxima l ideals,whic hw ehav e already encountered in section3 . It will benecessar yt oappl y thenotio no fprimenes st osubspace s ofa mor e general type thanideal . Inthi s sectionE i sa narbitrar y Riesz space.R(E)an d1(E ) are abbreviated toR andI respectively . 13.1. Definition. A linear subspaae L of E is said to be prime if it follows from x,ye E and xAye L that at least one of x and y is an element of L. 13.2. Example.I fE = C[0,1 ]an dL= ( xe E;x(0 )= x(l )= 0 }the nL isno tprim e becausei fx(t )= tande(t )= 1 fo ral lt e [0,1], the n x^(e- X)G L,howeve rx £ L an de - x f L. IfM i sth elinea r subspaceo fal l polynomialsi nE ,the nM i sprime , becausei fx, ye M the nxA ye M i fan donl yi fx an dyar e comparable, i.e.xA y=x o rxA y=y .I tfollow s thatx e M o ry e M.Not e thatM isno ta nidea lo fE . 13.3. Lemma. For a linear subspaae L of E the following assertions are equivalent. (a) L is a prime linear subspaae (b) if xVye L then xe L or ye L (a) if xVy= 0 then xe L or ye L (d) if xAy= 0 then xe L or ye L Proof: la)-*(b): xvye L implies -(xVy)= (-x)A(-y )€ L,henc ex e L o r Ye L . (b) •*• (o) :eviden t (o) ->• (d): xA y=0 implie s -(xAy)= (-x)V(-y )= 0 ,henc ex e L o ry e L (d) -*- (a) : if xAy = h e L then (x - h)A(y -h)=0, sox-heLor y - h e L, hence x e L or y e L. 40 Asa consequenc eo f Cd)w ehave :i ffo rlinea r subspaces Kan dLo fE holds that OL andL i sprime ,the n Ki sprime . IfX i sa subse to fE ,the nth eintersectio no fal l linear subspacesL ofE suc h thatL 3 X i sals oa linea r subspace,whic hw ecal l thelinea r subspaceo fE generate db yX ,i nformul a Lss(X). Completely similar,b yth m3-7 ,w eca ndefin eth eRies z subspace,res pectively ideal,band ,pola ro fE generate d byX a sth eintersectio n of all Riesz subspaces,respectivel y ideals,bands ,polar so fE whic h containX ,i nformul a Rss(X), respectively Id(X), Band(X), Polar(X). Inth efollowin gw eabbreviat e Rss(Xù{x})t oRss(X,x )an dId(Xu{x} )t o Id(X,x)fo rX c E ,x e E . Inthi s sectionw ear e especially interested inRss(X )fo ra give nX cE . In general,i ti sdifficul tt ogiv ea close d descriptiono fth e elements of Rss(X)fo rXc E arbitrary . Inth efollowin gw eshal l meet two exceptions toth erule ,th efirs ti n thm 13.4,wher ew econside r the Riesz subspace generated bya linea r subspace,th e second inth m13.2 2wher ew econside r the Riesz subspace generated bya subse t consistingo fa prim e Riesz subspacean da n arbitrary elemento fE . 13.4. Theorem. For a linear subspace L of E the Riesz subspace Rss(L) of E consists of all finite infima of alt elements of E which are finite suprema of elements of \., in formula Rss(L)= { A V x,,, ; x,|,e [ _ for all je J,k e K,wit h j'e J k e K JK JK Jan dK arbitrar y finite index sets}. Proof:I fR(L )i sth erigh t hand set,the ni ti seviden t that Rss(L)= > R(L). Wear e donei fR(L )i sa Ries z subspaceo fE . Ifx, ye R(L)the nw ehav e also xAye R(L) . By Bigard, Keimelan dWolfenstei n [1977,cor .1.2.15 ]w ehav e foral l finite indexsets I,Jan dKan delement sx ., •( ie I,je J ,k s K)tha t V A V x... •jk= 1A - V T x,• ê j °(i)ki ' ieljeJkGK w^wkeK this implies thati fx, ye R(L)the nw ehav e also xvye R(L) . A Further, if x= V x,-k and y= A V y,, jejkexJK i e I me M nm 41 thenx + y = A { V x..+ V y, } jeJ keK JK meM ™ ieI A v {x,k +y. }eR(L) . je J ke K JK im ie I me M IfA > O ,x e R(L)the nals oA xe R(|_). Finally,-( A V x,.)= V A (-x..) je J kG K JK jG j kG K JK = A J V (-"iahV£R(L) ' Itfollow s thatRss(L )= R(L). 13.5.Lemma . If xG E+then Rss({x})= {Xx ;X e ]R } Proof:eviden t 13.6. Theorem. For any Riesz subspaoe S of a Riesz space E and xG E such that x£ S , there exists a Riesz subspaoe R of Eu-ti/ ztfc e following properties (a) x£ R fW RD S ^C/)•*- ƒ for certain Riesz subspaoe T of E holds that X£ T an dT 3 R tfee w T= R . Proof:Le tR s= { UGR ;X £ U ,U 3 S )b epartiall y orderedb yinclusion , thenR s= £ If2 is'anarbitrar y chain in'ft^s ,the n Rss(US)e R andRss(lE ) oS . Ifx e Rss(US),the nx = A v x1L,fo rcertai n finite indexsetsJ jG J k G K jk and K ,x. . G U2 ;a sa consequenc eo fth efinit ecardinalit yo fth ese t {x..;j G J ,kG K }ther eexist sa W i nth echai nz suchtha t {x,k;j G J,k e K }c W ,s ox G W.Sinc ex £ W i tfollow stha tx £ Rss(U£). Hence,Rss(lE )G R Now,Zorn' s lemmaca nb eapplied ,an dth edesire d resultfollows . 13.7. Definition. A Riesz subspaoe R with the properties listed in thm 13.6 is called a Riesz subspaoe maximal in R with respect to the property 42 of containing S and not having x as element, abbreviated to R is S,x- maximal in R. A Riesz subspace R is called x-maximal inR if Rt sS,x - maximal inR /o rsom eS .R x s called anR-relative maximal Riesz subspace if Ri eS, x-maximal inR /or someS an< ix^ S . 13.8. Proposition. For Riesz subspaces R and S of E such that RD S and xG E such that X£ S i/z e following assertions are equivalent (a) Ri sS, x-maximal in R fW Ri s{0}jX-maa:ima Z•£ «R Proof:eviden t 13.9. Theorem, (cf.e.g .Luxembur gan dZaane n [1971, thm33. 5 ]). For any ideal J of a Riesz space E andx £ E such that x^ J j there exists an ideal Mof E with the following properties. (a) x£M (b) M3 J feji ffo r certain ideal No fE holds that x£ N an dN 3j j t/ze«N = J . 13.10.'Definition . An ideal Mwi£/ zt« e properties listed in thm13. 9 is called an ideal maximal in I with respect to the property of containing J and not having x as element, abbreviated to M is J,x-maximal inI . ^lw ideal M-i s called x-maximal in Ï if Mis J,x-maximal in Ifo rsom eJ . M is called an \-relative maximal ideal if M is J,x-maximal in \ for someJ and x£ J . 13.11.Proposition .Fo r ideals Man dJ o fE swc/ z that M3 J an d xe E SMC« that x£ J tn e following assertions are equivalent (a) Mi sJ,x-maxima Zi nI . (W Mi s{0 } ,x-maximal in \. Proof: evident InLuxembur gan dZaane n [1971,th m33. 4] f ti sprove d that every I-relativemaxima l ideal isprime . Herew egiv ea proo f basedo n the distributivityo fth elattic eI ,i nwhic hsu pan din far edenote db yV andA respectively . 13.12.Theorem . Every \-relative maximal ideal Mi s prime. Proof:i fM i sx-maxima l inI an dy, ze Ear esuc h that neithery no rz 43 isa nelemen to fM ,the n sinceM i sx-maxima l inI w e havex e Id(M,y) andx e Id(M,z),henc ex e Id(M,y)nid(M,z)= Id(M, I )nid(M,IE)= (MVI )A(MVI )= MV( I AIZ)= MV{0 }= M . Sincex $ Mi tfollow s thatM i s prime. In thm 6.12 the similarity between R-relative maximal Riesz subspaces and I-relativemaxima l ideals breaks down,because ,a sa consequenc eo f the fact thatR isno tdistributiv ei ngeneral ,fo ra Ries z subspacet o be primean dt ob eR-relativ emaxima l areindependen t properties.Thi s will beshow ni nth efollowin g example. 13.13. Example.Le tEb eth eRies z spacea sdefine d inex . 12.7.I fx e E is such that x(t)= t fo ral l te [0,1] ,an dS i sth eRies z subspaceo f Econsistin go fal ly e E suc h thaty(0 )= y(l) , thenS i sx-maxima li n R, becausei fS i sstrictl y containedi na Ries z subspaceR ,the n there existsa z € R suc h that z(0) + z(l). Nowfo rre E wit h r(t)= z(0)+(z(l )- z(0)) tw e haver - z e S ,henc er - z e R ,bu tthe n also re R ,henc es e R i fs(t )= (z(l )- z(0)) tfo r all te [0,1] , thu s (z(l)- z(0)) _1s= x e R . HoweverS i sno tprime ,becaus ei fe(t )= 1 fo ral l te [0,1] ,the n xA(e- x) e s,bu tx $ S,e - x £ S . IfW = { xe E;x(| )= x'(| )= 0} ,the n Wi sa prim e Rieszsubspace . ButW i sno tx-maxima l inR for somex e E. For,suppos e x'(J) *0 , then forT = { y€ E;y'(J )= 0 }hold s thatx £ T,bu tW i sstrictl y contained inT .I fx'(| )= 0 an dx(J )* 0 the n forV ={ y€ E;y(i)= 0 } holds thatx £ T,bu t Wi sstrictl y contained inV, henc eW i sno ta R-relative maximal Rieszsubspace . 13.14. Example.A nidea lJ in' a Riesz spaceE ca nb ex-maxima l inI fo r certainx e E withou t being x-maximal inR . IfE i sth eRies z subspace inth eRies z spaceo fal l real sequences,generate db yCQ Q (all real sequenceswhic har e eventually 0),e = (1,1,1,... )an dr = (1,2,3,...), then c„0 ise-maxima l inI ,bu te £ Rss(cQ0,r), hence CQ0 isno te - maximal inR . 13.15.Theorem . Every proper Riesz subspace R is the intersection of all Riesz subspaces S such that Rc S and S is ^-relative maximal. Proof: LetR R beth ese to fal l those Riesz subspacesS suc h thatR c s 44 n n andS i sR-relativ emaximal .I ti seviden t that Re Rn. Supposex e RR> x£ R. Then therei sa S e I Lsuc h thatS i sx-Taximali nR , hencex £ S , contradiction. Itfollow s thatR = n R We note thatth esituatio n inI i s completely similar. Often,tha t theo remi sgive ni na milde r form, namely that every proper ideal is the intersection ofal l prime ideals which contain it(cf .e.g . Luxemburg and Zaanen [1971,th m33. 5] ). Iti ssurprisin g thatjus tth eanalogu e of thismilde r form doesno thol d inR ,a sth efollowin g example shows. 13.16. Example.I fE = C [0, 1] an dR= { xe E ;x(0 )= x(l) } thenR i s a proper Riesz subspaceo fE .R i sno t prime,becaus ei fe(t )= 1 and y(t)= t fo ral l te [0,1] the ny A ( e- y )= 0 ,howeve ry £ R , e- y £ R .Th e only Riesz subspace which containsR properl y isE .Hence , the intersectiono fal l prime Riesz subspaces containing Ri sequa l to E. Iti swel l known (cf.e.g . Schaefer [1974,II I thm.2. 2] ) that (Me I;M 3 J }i sa chai ni nI wheneverJ i sa prim e ideal.Her e another disagreement appears;th efollowin g example will demonstrate this. 13.17.Example .Le tE b eth eRies z spacea sdefine d inex . 12.7.The n P= { xe E;x(-~ )= x'^ )= 0 }i sa prim e Riesz subspaceo fE .Th eRies z 1 1 subspacesR = { xe E;X(^ )= 0 }an dS = { xe E;x'(p- )= 0 }bot h contain P strictly,bu tR ? S an dS 13.18.Definition . A prime Riesz subspace R is oalled minimal in R with respect to a Riesz subspace S if RD S and if it follows from Sc Tc R for certain prime Riesz subspace T that T= R . We abbreviate this by saying that R is a prime Riesz subspace which is S-minimal in R. R is called ^-relative minimal if R is S-minimal in R for someS . 13.19. Example.I fa prim e Riesz subspacei sR-minima l inR for some Riesz subspace Rthe ni ti sno tnecessaril y{0}-minima l inR becausei f E= C [0, 1 ]an dR i sa si nexampl e 13.16 theni tfollow s from this example thatE i sR-minima l inR . ButE i sno t{0}-minima l inR , becausefo r J= { xe E ; x(0)= 0 }i thold s thatJ i sa prim e ideal andJ^ E. 45 13.20.Theorem . For every Riesz subspaae R there exists a prime Riesz subspace S which is R-minimal in R. Proof: LetR R ={ T€ R ;T = >R an dT prime }b eordere db yanti-inclusion . RR* 4,, becauseE e R R.Le ts b ea chai n inR R. Thenb yth m3. 7nseR , alsons D R. IfxA y=0 fo rx, ye E an dx£ n s,y f nithe n there exist K,L e s such thatx £ K,y # L.Withou t losso fgeneralit yw ema y suppose KCL. Thenx £ Kan dy£ K .Henc e n2 isprime .I tfollow s that n 2 isa nuppe r boundo f2 i nR, .B yZorn' s lemma there existsa Ries z subspace which isR-minima l inR . Inth eligh to fexampl e 13.16w ecanno t expect that every Riesz subspace Ri sth eintersectio no fal lRies z subspaces whichar eR-minima l inR , although iti scorrec ti fR i sa nideal ,a sw eprov ei nth efollowin g theorem. 13.21.Theorem . Every ideal J is equal to the intersection of all prime Riesz subspaces which are à-minimal in R. In particular, the intersection of all prime Riesz subspaces which are {^-minimal in R is equal to {0}. Proof:B yLuxembur gan dZaane n [1971,th m33. 6] J i sequa l toth einter sectiono fal lprim e idealsM suc h thatM = >J .Fo r every prime idealM such thatM o Jther e existsa prim e Riesz subspaceR suc h thatR cM andR i sJ-minima l inR . Butthe n certainlyJ i sequa l toth eintersectio n ofal lprim e Riesz subspaces whichar eJ-minima l inR . 13.22.Theorem . For every prime Riesz subspace R and all x£ E we have that Rss(R,x)= Lss(R,x )= {r+ Ax ;r e R ,A eR } Proof:Le tK= { r+ Ax ;r e R ,A GP} . From Rss(R,x)3 K i tfollow s that iti ssufficien tt oprov e thatK i salread ya Ries z subspaceo fE . Thereforew eshal l prove that (r + Xx)V( s+ ux) eK fo rr, se R an d A,ye ]R .R i sa prim e Riesz subspace, (s- r + ( u- A)x )A( r-s + ( A- y)x ) =0 ,henc e (s- r + ( u- A)x) + eR o r( r- s + ( A- y)x) +e R . Sincer, se R i tfollow s that (s- r + ( u- A)x ) +r e R o r( r- s + ( A- y)x ) +s e R ,henc e (s+ ( y- A)x)V r G Ro r( r+ ( A- y)x)V se R .Thi s implies that (s+ ux)V( r+ Ax )- A xe R o r( r+ Ax)V( s+ yx )- y xe R ,hence , there exist t,uG Rsuc h that 46 (r+ Xx)V( s+ yx )= t + X xo r( r+ Xx)V( s+ yx )= u + yx ,s o (r+ Xx)V( s+ yx) eK . 13.23. Definition. A Riesz subspace R is called strongly compact in R if it follows from R= n r where Fc R that R= S /o rsom eS e r . 13.24.Theorem . A Riesz subspace is ^-relative maximal if and only if it is strongly compact in R. Proof:I fR i sa Ries z subspace,x-maxima l inR andP i sa collectio n of Riesz subspaces such thatR = nr , then there existsa S e r suc h thatx £ S.Togethe rwit hS = >R thi s impliesS =R . Conversely,i fR i sa strongl y compact Riesz subspace,the n letR *b eth eRies z subspacewhic h isth eintersectio no fal l Riesz subspacesS suc h that Res properly. ThenR *als o properly containsR ,becaus eR *= R woul d imply thatR isequa l toa Ries z subspacewhic h properly containsR ,a contradiction . Ifx G R*\Ri sarbitrary ,the nR i sx-maximal ,becaus ei fS = >R , and S * R,the nth eintersectio n isals o taken over S,henc ex e S becaus e xe R* . 13.25.Theorem . A prime Riesz subspace R=* =E is ^-relative maximal if and only if there is a xG E such that Rss(R,x)= E . Proof: SupposeR i sR-relativ emaxima l andther eexist sn ox s Esuc h that Rss(R,x)= E. Lety e E\Rb earbitrary .The n Rss(R,y)# E,henc e there existsa z eE\ R such thatz f Rss(R,y). Byth eforegoin g theoremw ehav e Rss(R,y)= { r+ Xy ;r G R,X eR} , Becausez £ Rss(R.y)w ehav ez + r+ X yfo ral lr e R an dXG ]R . y e Lss(R,z)woul d implyy =s + y zfo rcertai ns e R an dy G R ,y *0 becausey £ R ,henc ez = y "( s- y) , soz e Lss(R,y),contradiction , hencey f Lss(R,z),s ob yth eforegoin g theoremy £ Rss(R,z). Itfollow sno wtha tn{Rss(R,y) ;y S P.}= R ;b yth estron g compactnesso f Rw ehav eno wR= Rss(R,y )fo rsom ey £ R ,contradiction . Conversely,i fRss(R,x )= E fo rsom ex e E an dRes , R= *S for m some Riesz subspaceS ,the nfo ra narbitrar yy e S\ Rw ehav e that y e Rss(R,x)= Lss(R.x) ,s oy = r + X xfo rsom er e R an dX = £0 . But thenx e Lss(R,y)= Rss(R,y )c S ,henc ex e s.I tfollow s thatR i s 47 x-maximal inR . 13.26.Theorem . If R is a prime Riesz subspaoe which is ^.-relative maximal, then Lss(R,y)= E for every y eE\R . Proof:B yth eforegoin g theorem there existsa nx e Esuc h that Rss(R,x)= E. Lety e E\ Rb earbitrary ,the n there existsa A # 0 suc h thaty = r +A x forsom er e R .Henc ex = - A r+ A ~y .I fz e E the nz = t + ux ,wher e te R ,s oz = ( t- A _1yr)+ A _1yye Lss(R.y). Itfollow s that Lss(R.y)= E. 13.27.Definition . A Riesz subspaoe R is called (absolute) maximal in R if the only Riesz subspaoe which contains R properly is E. Iti seviden t thatever y Riesz sunspacewhic h is (absolute)maxima l iftn is R « relativemaximal .Th econvers e holds underth eadditiona l assumption thatth eRies z subspacei sprime ,a sw esho wi nth efollowin g theorem. 13.28.Theorem . Every prime Riesz subspaoe which is ^-relative maximal is (absolute) maximal in R. Proof:thi si sa nimmediat e consequenceo fthm .13.26 . 13.29.Theorem . If R is a prime Riesz subspaoe which is ^-maximal in R for some x€ E, then for every y€ E there exists precisely one A G] R such that y -A x€ R. Proof:B yth m13.2 6w ehav e Lss(R,x)= E . Ify e E the ny = r + A x forcertai nr G R,A eR , hencey - A x eR . Ifals oy - A xe R fo r certain Ae]R,sa yy - A x= s ,the ns - r = (A -A) xe R .Fro mx f Ri tfollow sno wtha tA = A . Inth efollowin g theoremw edetermin eth enullspace so fdisjunctiv e linear functionals o n E. 13.30.Theorem . The nullspaces of non-trivial disjunctive linear funo- tionals are precisely the prime Riesz subspaces which are ^-relative maximal. Proof: Let T be a non-trivial disjunctive linear functional on E and 48 denoteth enullspac e N(T)o fT b yN . xe N implie s |TX|= 0 ,henc e |T|X||= 0 b yth m12.2 ,henc e |x|G N . Togetherwit hN a linea rsubspac ethi s implies thatN i sa Ries z sub- space. Furthermore,N i sprime ,becaus e ifxA y=0 the nTx lTy ,henc eT x=0 orT y= 0 ,s ox e N o ry e N . BecauseT i snon-trivia l thereexist s axe E\Nsuc h thatT x=1 . Noww ehav etha tN i sx-maxima l inR , because,i fR 3 N an dR ^N the n thereexist sa y e R wit hT y= 1 .No wx- y G N,henc ex - y e R, but thenals ox e R . Conversely,le tNb ea prim eRies zsubspac ewhic hi sx-maxima l inR . Byth m13.2 9ther eexist sa ma pT fromE t oR ,whic h assignst oy e E the real numberA such thaty - A x eN . Weshal l proveno wtha tT isa disjunctiv e linearfunctiona l withnull - spaceN .T islinea rbecaus ei fy, zG Ethe ny + z - ( A +A ) xGN , henceT ( y+ z )=T y+ T zan dfo rAe ] Rw ehav eA y- A Ax G N,henc e x x x y Tx(Ay)= AT xy_ Fory e E w ehav eb ydefinitio ny - ( Ty) x G Nan d|y |- ( T|y|) xG N , hence + |y| - (Tx|y|)x + (y - (Txy)x) = 2y - (Tjy| + Txy)x e N and |y|- (T x|y|)x- ( y- (T xy)x)= 2y "- (Tjy |- Vyj x GN . 2yA2 y =0 ,henc e 2y+G N o r2y ~e N becaus eN i sprime . Itfollow sno wtha tals o (T |y|+ T y) x G No r( T|y |- T y) xG N .B yth efac ttha tx ? N w ehav e XX XX Tx|y|+ T xy= 0 o rTjy |- T xy= 0 ,henc e(i nbot hcases ) |Tx|y||= |Txy|, soT isa disjunctiv e linearfunctional . N(T) =N becaus eT y = 0 fo ry G E impliesy = y - O xG N an dyG N X X impliesy - O xG N,henc eT y = 0 . 13.31.Theorem . For every 0= £x G E there exists an x-maximal prime Riesz subspace. Proof:I tfollow s fromth m13. 9tha tther eexist sa nidea lJ maxima li n I with respectt oth epropert yJ ^ {0 }an dx£ J .J i sprim eb yth m13.12 . Nowb yth m13. 6ther eexist sa Ries zsubspac eR maxima l inR wit h respect toth epropert yR = >Jan dx f R. FromR 3 J an dJ prim ei tfollow s thatR i sprime . 49 13.32.Corollary . For every Riesz space E# {0} the intersection of all prime Riesz subspaces which are n-relative maximal is equal to {0}. 13.33.Theorem . The set of all disjunctive linear functionals on an arbitrary Riesz space E is total, i.e. not empty and such that x= 0 if Tx= 0 for all disjunctive linear functionals T onE . Proof: From thm 13.31an dth m13.3 0i tfollow s thatth ese to fal ldis junctive linear functionals isno tempty . IfT x= 0 fo ral ldisjunctiv e linear functionalsT o nE the nx i sa nelemen to fth eintersectio no f all nullspaceso fdisjunctiv e linear functionals,hence ,b yth m 13.30 and cor. 13.32w ehav ex = 0 . Itwa sconjecture d byKrul l thatfo rever y Archimedean Riesz spaceE there existsa ninjectiv e Riesz homomorphismT fro mE t oa Riesz space ofall .realvalue d functionso na nappropriat enon-empt y setS .A counter examplet othi s conjecturewa sgive nb yJaffar d [1955/56] an dman y S others. Note thati fE i sa nArchimedea n Riesz spacean dT :E'-* K is an injective Riesz homomorphism, then cj>°Ti sa realvalue d Riesz homo- morphismfo rever ypoint-evaluatio ni n] R, henc ei nE ther e exist maxi mal ideals.Bu tther e exist Archimedean Riesz spaces withoutan ymaxi mal ideals (cf.e.g .Schaefe r [1974 , III,exc.6']) . A general representation theory forArchimedea n Riesz spaceswa sgive n by Bernau [1965 a ], who .prove d that every Archimedean Riesz spaceca nb eimbedde d (bya ninjectiv e Riesz homomorphism)i na Ries z spaceo fextende d realvalued continuous functionso na nappropriat e compact Hausdorffspace . Relatedt othi si sa theore mo fBrow nan dNakan o [1966] whic h states that every Archimedean Riesz spacei sa facto r spaceo fa nappropriat e S Riesz spaceR . Withth e helpo fnullspace so fdisjunctiv e linear functionalsw eca n proveth eKrull-typ e conjecture foraT disjunctiv e linear operatorin steado fa Ries z homomorphism. 13.34.Theorem . For every Archimedean Riesz space E there exists an injective disjunctive linear operator T from E to a Riesz space of all real valued functions on an appropriate non-empty set S. Proof:Le tV b eth ese to fal ldisjunctiv e linear functionalso nE an d 50 letF b eth eRies z spaceo fal l realvalued functionso nP wit h pointwise linear operationsan dpointwis e partial ordering. Defineth elinea roperato rT :E -» •F b y(Tx)(S )= S xfo ral l Sep. xl y fo rx, ye E implie sS xl S yfo ral l Sep, henceT xi T yi nF , so T isa disjunctiv e linearoperator . IfT x= 0 fo ral l xe E the nS x= 0 fo r allS e P ,henc eb yth m 13.33 we have thatx = 0 ,s oT i sinjective . 14. Exampleso funbounde d stabilizers Inthi s section wegiv esom ene wexample s ofunbounde d stabi lizersan dw egiv e some counterexamplesfo r stabilizerso fstatement s which hold fororthomorphisms . 14.1. Example.Le tEb eth eArchimedea n Riesz spaceo fal l functions xo n[1, 2] whic har epiecewis e polynomials,i.e .t oever yx e E ther e t x exist real numbers x-(x)suc h that1 = T„(X)< .. .< n+i( ) =2 suc h thatx coincide s witha polynomia l x-o never y [T.(X), T-+,(X))fo r T i= 0,...,n- lan da polynomia l x on[ T(x) ,n+i(x) ]• Toan yt e [1,2)an dan yx e E ther ei sa ni(t,x )suc h that 1G [T i(t,x)(x)' Ti(t,x)+l(x))'1e t ^2jX)= n -The n define Tt: E-FbyTtx = xi(tfX)(0). Ifw edefin ea rin g structureo nE b ypointwis e multiplication,the n foral lt e [1, 2 ]th eoperato rT .i sa rin g homomorphism such that T.e= 1 (wher ee i sth efunctio n constant1 o n[1, 2] an dN(T. )= J. , whereJ .i sth e maximal ring ideal ofal l functionsx suc h that x i(t5x)(°> =°- ButT .i sfo rte [1, 2 ]no tonl ya rin g homomorphism,bu tals oa dis junctive linear functional,becaus ei fx ly i nE the n x..,. x\(0)= 0 ory.j, . JO) =0 ,henc eT. x= 0 o rT. y= 0 ,s oT txiT ty. Noww edefin ea linea roperato rT fro mE t oitsel fb y(Tx)(t )= T.x( te [1,2]]). Ifxl yi nE the n x(t)= 0 o ry(t )= 0 fo ral l te [1, 2 ], s oi f x. ±0 51 for some i= 0,..., n theny i(t>y)(t)= 0 for te [Ti(t>x)(x), T.(t>x)+1(x)), so xiTy,henc eT isa stabilizer on E. T is unbounded, because T([0,e ]) i s unbounded in E.Namely , iff e seq(E) is such that (f(n))(t)= -n t + n+ n" 1 ift e [1, 1 + n"2 ] and (f(n))(t)= 0 ift e [1 + n" ,2 ] ,the n for all n> 2w e have 0< f(n)< e, however (T(f(n));n > 2} is unbounded,becaus e T(f(n))(0) =n + n-1 for all ne N. T isno t regular continuous,becaus e foral l neH we have 0< f(n)< n e, hence Lf= 0 ,bu tT f is noteve n order convergent,becaus e by thefore going {T(f(n));n > 2} is unbounded. HenceT is also not order continuous. Another example of an unbounded stabilizer on Ei s the following. Iffo r every te [ 1,2 ) and xG Ew ewrit e x'... >(t)fo r the right deri- 1[X,x) vate of the polynomial x. f. . in the point tan d x.,9 VJ2) for the left 1[ X , X J 1^* _, XJ limit'o fx'., . vJt) for t -*2 ,the n define forever y te [1,2 ]a linea r operator S. from Et oF byS. x =x'., . x^(t). Now again S. isa disjunctiv e linear functional forever y te [1, 2] . Define alinea r operator S from Et o itself by (Sx)(t)= S x (te [1,2] ) , thenwit h asimila r proof as forT we can show thatS isa stabilizer; with the same order interval [0,e]an d the same fe seq (E)w e can show that alsoS is unbounded. Alsowit h the same fe seq (E)w e can show that S isno t regulator continuous and notorde r continuous. Ifw e compare s andT , then the following differences are conspicuous. ? We haveT e =e ,bu tS e =0 ,furthe rT =T , but fors holds that for nx every xe Ether e exists an e Msuc h thats x= 0 (take n the maximum X X of all degrees of the polynomials x- (i = 0,..,n)) . Ingeneral ,orthomorphismso n an Archimedean Riesz space commute (Bernau [1979 ]). However,arbitrar y stabilizers on an Archimedean Riesz space need notcommute . We give an example in the above Riesz space E. LetR be the orthomorphism on Ewhic h is the rightmultiplicatio n with z where z(t)= t foral l te [1,2] . ThenR S * SR because RSz =R e =z ,bu tSR z =S z =2z . 52 ChapterI V f-ALGEBRAS In this chapter Riesz spacesar estudie d which also havea rin g structure, which isconnecte d withth eRies z space structurevi aa ver y strong com- pability condition,th esocalle d f-algebras. Relations between ring ideals and (order) idealsar ediscussed ,jus ta srelation s between ring homo- morphismsan dRies z homomorphisms.A varian ti sgive no fa theore mo f Ellis[ 196 4] an dPhelp s [196 3] . 15. Introductiont of-algebra s f-algebras were introduced inNakan o[195 0] ,Amemiy a[195 3] an dBirkhof f and Pierce[ 195 6] .Befor e giving thedefinitio no ff-algebra ,w egiv eth e definitiono fRies z algebra,whic h isles s restrictive. 15.1. Definition. A Riesz algebra is a quadrupel (E,+ ,< ,. ) where (E,+ ,< ) is a Riesz space and(E ,+ ,-.- ) is an algebra, such that x .y è E for all x,y e E. Inth eseque l forfixe d x,< an d .w eabbreviat e (E,+ ,< , .)t oE . Unless otherwise statedw eindicat ea produc to ftw o elementso fa Ries z algebrab yjuxtaposition . 15.2. Definition. Anf-algebra is a Riesz algebra E in which for every x, y, z e E holds that y A z = 0 implies xy A z = 0 and yx A z = 0. 15.3. Theorem, (cf. Bernau [1965b ] for positive elements) A Riesz algebra E is an f-algebra if and only if for all x, y e E holds that xy e r n y , Proof: +: if x, ye E then x A z = 0 for some z e E implies xy A z = 0, hence xy e x . Likewise xy £ y . If x, ye E are arbitrary, then by the x • u + + ,- / +\J"L + - ,. , +J.1 - + ^ , -.11 . foregoing we have xye(x) , x y G (x ) , x y G (x ) and x~y" e (x ) .It follows that xy = (x+ - x")(y+ - y") = x y - x y - x"y + x~y~ e (x ) + (x~) c |x| = x . Likewise xy e y » hence xye x n y . -t-: i fy A z = 0 fo rsom ez e E,the nfo ral l xe E wehav e 0 53 15.4. Definition. An f—multiplication on a Riesz space (E,+ ,< ) is a mapping . :E x E + E such that (E,+ ,< , .) is an f-algebra. We note that every Riesz spaceca nb egive nth estructur eo fa nf-algebra , namelyb yth e zero-multiplication. Although this f-multipl icatio n seems tob eo fn ovalu ea ta firs t glance,sometime s iti simportan t however; fora nexampl ese eKeime ![1971 , Introduction ].Suc ha nf-algebr ai s calleda zero-algebra . Inou r case f-algebraswit h thepropert y that many products vanishca n give riset opathology .A simpl e extra assumption canavoi d this.W e call f-algebras with this property faithful. They will beintroduce di n the following section. 15.5. Definition. For an element x of an f—algebra ¥, left multiplication by x is denoted by £ , and right multiplication by (R. 15.6. Lemma. For each f-algebra F and each xe F the mappings £ and Ü are orthomorphisms on F. Moreover, they are positive in the operator ordering if x >0 . Proof: straightforward. 15.7. Examples. (a) ]R with componentwise linear operations, componentwise partial or dering and componentwise multiplication. Note that there exists a multiplicative unit, namely e = (1,1). (b) The componentwise multiplication on (]R , lex) is not an f-multipli- cation because (1,-1) > 0, (0,1) > 0, but (1,-1)(0,1) = (0,-1) ^ 0. Note that (x, ,x2)(yi ,y2) = (x-,y-, jX^y-^) is an f-multiplication on (F ,lex) ,bu ti sno tcommutativ e (c)Th eRies z space C(X)wher eX i sa topologica l space,provide d with pointwise multiplication isa nf-algebra . (d)Th eRies z space cQ0o fal l sequenceso frea l numbers whichar e eventually zero,provide d with componentwise multiplication is an f-algebra. 15.8. Remark. Not every Archimedean f-algebra witha stron g order unit is Riesz isomorphic toa Ries z space C(X)wit hX a (pseudocompact) 54 Hausdorff space (fora definito no fpseudocompac tse eGillma nan dJeriso n [1976,1. 4] ). Ifth eRies zspac eF Ro fal l sequenceso frea l numberswit h a finite rangei sgive na nf-multiplicatio n structureb ycomponentwis e multiplication,the nF Ri sa nArchimedea n f-algebrawit h strong order unite = (1,1,1 , ...). HoweverF Ri sno tregulato rcomplete ,becaus eF R hasP Pbu ti sno tDedekin d complete (Aliprantisan dBurkinsha w[1978 , Ex. 2.13(3)] an dth m 7.9). HenceF Ri sno tRies z isomorphict oan yRies z spaceC(X )b yth m7.10 . 15.9.Theorem . In every f—algebra F we have for all x,y ,z e F that (a) z> 0 implies z(xV y )= z xV z y and( xV y) z= x zVyz . (b) |xy|= |x||y| . Ca)x 1 y implies xy=0 . Cd)x 2>0 . (e) 0.< z < x andy x >0 implies yz> 0 (y not neoessavily positive). (f) if x,y > 0 , xy =yx and( xV y)( xA y )= ( xA y)( xV y ) then (xA y)c = xL A yc, (xV y)c = xc V yc . (g) if xy =yx, x~y =y x and (xV y)( xA y )= ( xA y)( xV y ) then (xV y)( xA y )= xy. Proof: (-a), (b), (a) and(d) aregive nb yBigard ,Keime lan dWolfenstei n [1977,prop.9.1.1 0] . Ce)y x> 0 ,s oy x - y" x> 0 .Togethe rwit hy x A y" x= 0 thi s implies thaty" x= 0 .Bu tthe n alsoy~ z= 0 ,henc e yz=yz-yz=yz>0. (f) (xV y)( xA y )= ( xV y) xA ( xV y)y =(x 2V yx )A (x yVy 2) 2 2 2 2 =( x vxy )A (x yVy ) = ( x Ay ) V x y> xy .O nth eothe rhan dw e have( xV y)( xA y )= ( xA y)( xV y )= ( xAy) xV ( xA y)y = (x2A yx )V (x yA y 2)= (x 2A xy )V (x yA y 2)= (x 2V y 2)A x y A remarkable propertyo fArchimedea n f-algebrasi sth e following theorem, forwhic h there exist twoentirel y different proofs,on eb y Zaanen[197 5] bymean so f thm 12.20.Anothe r proofi sgive nb yBerna u[ 1965 b] . 15.10.Theorem . In every Archimedean f-algebra the multiplication is commutative. The following theorem shows thatf-multiplicatio ni na nArchimedea n f-algebra issequentia l continuouswit h respectt oeac ho f thethre e typeso fconvergence ,define di nChapte r II. Iff ,g e seq(F)fo ra nf-algebr a F,the n the sequenceh wit h h(n)= f(n )g(n )fo r alln e IM isdenote db yfg .I fxe F the n xf, fxe seq(F)ar edefine db y(xf)(n )= xf(n) , (fx)(n)= f(n) x foral ln e h. 15.11.Theorem . If f and g are sequences in an Archimedean f-algebra F and (a) Lf= x , Lg= y then Lfg =x y (b) rLf =x ,r Lg= y then rLfg= x y c c c (c) Lf= x , Lg= y then Lfg= x y Proof: (a)i f p,qe se q (F)ar e such that |f- x |< p 4 -0 and |g-y |< q + 0 ,the n |fg - xy| < |fg - fy| + |fy - xy| = |f||g- yI + If- x||y |< (| f- x |+ |x|)| g-y |+ | f- x||y | < (p(l)+ |x|)q+ p|y| . z .y. |Ian fi.d Iar eorthomorphism si nF ,henc eb yth m 12.15se quential order continuous.I tfollow s that (p(l)+ |xj) q+ p|y | \0 , hence Lfg= xy . (b)i fr L(f,u)= x ,r L(g,w)=y fo ru ,w e F , the n for every e> 0 ther e existsa NN ee u ue sucsuchh tha thatt fof or al l nefl hold s that |f(n)- x |< E U and |g(n)-y| 56 LetO < e< 1 ,the nw ehav efo ral ln e ]\| withn > max(N,, N) tha t |f(n)g(n )- xy |< |f(n)g(n )- f(n)y |+ |f(n) y- xy| < (|f(n)-x |+ |x| ) |g(n)- y |+ |f(n )- x||y| < (u + |x|) ew+ eu|y| = e(u+ |x|w+ u|y|), hence rLfg = xy. (c) if for all ne M wehav e that P andQ n are polars in F such that f(n) - x e Pn, g(n) - y e Qn for all ne n andP p+1 c Pn, Qn+1 c -Qn for all ne M andn {Pn; n e ]N} =n {Qn; n e u } = {0}, then for all ne n we have that |f(n) g(n) - xy| < |f(n) g(n) - f(n)y| il + |f(n)y - xy| = |f(n)||g(n) - y| + |f(n) - x||y | eMP^c (Pp + QJ by thm 15.3,henc efo ral ln e M wehav e that f(n)g(n )- x ye ( p+ Q )li , n +yQ n With lemma8. 1an dth m8. 2i tfollow sno wtha t cLfg= xy. 16. Faithful f-algebras f-algebras forwhic h also holdsth econvers eo fth m15.9(c),th e socalled faithful f-algebras,ar einteresting . They havealread y been studiedb y Bigard,Keime lan dWolfenstei n [197 7] underth enam e "f-anneau réduit" andb yCristesc u [1976,4.3.3. ] underth enam e "normal lattice ordered algebra".I nth elatte rth eunderlyin g Riesz spacei ssuppose d tob e Dedekind a-complete. Herew egiv e anotheraccount . 16.1. Definition.An f-algebra Ffor which holds that xy= 0 impliesxl y (x,ye F )is called a faithful f-algebra. 16.2. Theorem. For an Archimedean f-algebra F the following statements are equivalent (a)F is faithful 2 (b)x >0 implies x >0 inF (a) the nilradical of Fis equal to{0} . Proof: 2 (a)-*(b) x =0 implie s xix,s ox= 0 (b)^-(c) ifn e ] Ni sarbitrar yan dx * 0the n |x|> 0 ; repeated application of (b)give s |x|2n> 0 ,henc e x2" * 0,bu tthe n certainly xn *0 . (c)^-(a) iti ssufficien t toprov e thatfo rx, y >0 i nF hold s that xAy>0 2 impliesx y> 0 . xAy> 0 implie s (xAy) >0 ,bu tthe ni ti ssur e that (xAy)(xAy)> 0 ,hence ,b yth m15.9(g) ,x y>0 . 57 16.3.Lemma ,Jƒ x and y are positive elements of a faithful Archimedean 2 2 f-algebra F, then x >y holds if and only if x >y ./I sa consequence 2 2 x =y holds if and only if x= y . Proof: Ifx >y ,the nx - y > 0.Togethe rwit h x> 0 an dy > 0thi s 2 2 2 2 impliesx -x y > 0,s ox >x yan dx y- y >0 ,s ox y> y .No ww ehav e 2. 2 x >y . 2 2 2 Conversely,i fx >y ,the nw ehav e (xVy- x ) < (xVy- x)(xV y+ x )= ? 2 (xVy)^ -x . 2 2 2 By thm 15.9 fƒ ) we have(xVy) = x Vy , and this implies / > ,2 . 2W 2 2 2 2 _ (xVy -x) Fromno wo nth esymbo l e (ore p)i salway s reserved forth emultiplicativ e unito fa $-algebr aF . The following lemmawil l beuse d frequently. 16.5.Lemma . If x and y are arbitrary elements of an f-algebra F and 2 2 y> 0 then for every ne f jwe have xy - nx y+ n y > 0aw d 2 2 xy - ny x+ n y > 0. 2 Proof:i fF i sa $-algebra ,the nb yth m15.9( W wehav e (x- ne ) >0 , 2 2 hence (x- ne )y > 0an dy( x- ne ) >0 ,bu tthe n certainly 2 2 2 2 xy - nx y+ n y >0 an dy x -ny x+ n y > 0 . If F is an arbitrary f-algebra, we have (ny - xy) A(x y - ny) = 0, so 12+ + by definition (xy - -x y) A(xy - ny) = 0, which implies [ (xy - ^x2y)A(xy - ny) ]+ = 0, so [ xy - 4x2yAny) ]+ = 0, but then 12+ 2 2 certainly (xy - —x y - ny) =0, hence x y - nxy + n y > 0. The following theorem is proved by Bigard, Keimel and Wolfenstein [1977, cor. 12.3.9 ] . Here we give a direct proof. 16.6. Theorem. Every Archimedean ^-algebra F is faithful. 2 Proof:b yth m16. 2i ti ssufficien tt oprov e thatx >0 fo rever y 0< x e F . 58 IfO < x G Fthe n sinceF i sArchimedea n thereexist sa n e I Nsuc h that + 2 (nx- e ) >0 .B yth m 15.9(d) wehav e(n x- e ) > 0,henc e (-n2x2+ 2n x- e) + =0 .No w(n 2x2- nx) + =(n 2x2- nx) ++ (-n 2x2+ 2n x- e)+ 22 22 + + > (nx -n x- nx +2n x- e ) =(n x- e ) >0 . 2 + 2 So(n x -x ) >0 an dthi s impliesx >0 . 16.7. Lemma. The multiplicative unit e in a ^-algebra ? is a weak order unit. 2 Proof:e >0 becaus ee = e >0 . eAx= 0 implie se x= x = 0 ,henc ee i sa wea k orderunit . Although every Riesz spaceca nb egive na nf-multiplicatio n structure, therear eRies z spaces,eve n with weak order units,whic h cannotb e givena ^-multiplicatio n structure.W ecom e backt othi s questioni n ChapterVI . InArchimedea n $-algebrasals oth econvers eo flemm a 15.6holds . In fact theyar echaracterize d bythi s property. 16.8.Theorem , (cf.Zaane n [1975,th m3 ]) . If T is an orthomorphism on an Archimedean ^-algebra F^ then there exists a zG F such that T =£ . 11 Proof:i fz = Te ,the nT an d £ coincideo n{e} .Becaus ee =F w ehav e T= J C byth m12.20 . Iti sa direc t consequenceo fthi s theorem thatz i sth euniqu e element ofF suc h thatT = £ ,becaus ei fals oT = £ ,fo r z'€ F ,the n £z _z ,"'= £z - £z, =0 ,henc e (z- z') e= 0 ,s oz = z' . IfT > 0 ,the nz > 0 becaus ez = Te . 16.9. Theorem. For every Archimedean ^-algebra F it holds that F is f-algebra isomorphic to 0rth(F). Proof: ifA i sth e mapping fromF t o0rth(F )whic h assignst oz the element £ of0rth(F) ,the nA i sa linea roperator ,bijectiv eb yth m 16.8an dlemm a 15.6. A ispositiv eb yth eforegoing . Itfollow s from Luxemburgan dZaane n [1971,th m18. 5] thatT i sa Ries z isomorphism. Furtherfo ral lx, ye F w ehav eA(xy )= £ =£ £ =(Ax)(Ay) ,sinc e xy xy multiplicationi n0rth(F )i scomposition . Hence,T i sals oa rin g isomorphism. 59 17. Idealsan drin g ideals From the last theoremo fth e foregoing sectiona fundamenta l relation canb ederive d between orthomorphismso na nArchimedea n $-algebraF andth e ^-multiplication structureo n F. Weus eth ewor d ideal for order ideal andrin g ideal fora twoside d ringideal. 17.1. Theorem. Every orthomorphism on an Archimedean ^-algebra F preser ves all ring ideals of F. Proof:i fT e Orth(F), thenb yth m 16.8ther e existsa z e F suc h that T =X . z Forever y ring idealJ o fFw ehav e nowT x= z xe J fo ral lx e J . Alsoi nfaithfu l Archimedean f-algebrasan deve ni na certai n sensei n every Archimedean Riesz space this phenomenon playsa nimportan t role. Iti swel l known (cf.e.g . Birkhoff [1967, XVII.5,theore mo fFuch s] ) thata Ries z algebra isa nf-algebr ai fan d onlyi fever y polari s a ring ideal.I na nArchimedea n f-algebra even every o-ideal isa rin g ideal. 17.2.Theorem . Every o-ideal J of an Archimedean f-algebra F is a ring ideal. Proof:i ti ssufficien tt osho w that0 < x e J an d0< y G Fimplie s xye J .I ffe se q(I v)i ssuc h that f(n)= xyAn x for all ne fi then il 0< f an dtf .B yth m15. 3w ehav ex ye x ;fro m Luxemburgan dZaane n [1971,th m 24.7] i tfollows'no w that ftxy,henc eb yth m 6.1(a) xy= °Lf , sox ye J . 17.3. Theorem. Every r-ideal J in an f-algebra F is a ring ideal. Proof: it is sufficient to Drove that 0 < x e j and 0 < y e F implies 1 2 xy e J and yx e j. By lemma 16.5 we have yx - nx < -y x and thus + 12 (yx - nx) < —y x for all n e IN. Define f e seq (I ) by f(n) = yxAnx for all n e n, then 0 < f and tf. * +12 For all n e ]N we have now |f(n) - yx| = |0A(nx - yx)| = (yx - nx) < —y x, hence yx e Lf, so yx e J. Itca nb eprove d similarly thatx ye J; thenx y canb etake na sa regulator . 60 17.4. Theorem. Every o-ideal J in an Archimedean f-algebra Y is a ring ideal. Proof:i ti ssufficien tt osho w that0 < x e J an d0< y G Fimplie s yxe J . Iff e se q( I) i ssuc h that f(n)= yxAn x foral ln e IN , then0 < f an d tf. Noww ehav e that f(n)- y x= 0A(n x- yx )= -(y x- nx ) G (yx- nx) + +11 Foral ln e ] Ni thold s that(y x- ( n+ l)x ) c (yx- nx) Iffo r0 < v e F hold s thatv G (yx- nx ) thenvl(y x- nx ) foral l ne ]N . SinceF i sArchimedea nw ehav ey = sup{( y —yx) ;n e IN},b y thm3. 5i tfollow sno wtha tvly , hencevlyx ,bu tthe nfo r all nef)w e havevl(y x- nx ), henc ev G (yx- nx ) .I tfollow s thatv =0 , so n{(yx- nx) +ii; ne N } ={0} .Henc ey x= °Lf ,s oy xe J . 17.5. Remark. Notever y idealJ i na f-algebr aF i sa rin g ideal.I fF isth ef-algebr a ofal l sequences ofrea lnumber swher eth e^multi plicationi sth ecomponentwis e multiplication,an dJ i sth eidea lo f all sequences convergingt o0 ,e = (1,1,1,... )G F an dy = (1,^,-j,.. )e J , z= (1,2,3,... )G Fthe ny z= e£J . In some f-algebrasw ear ei nth epleasan t situation that every ideali s a ring ideal.B yLuxembur gan dZaane n [1971,p.210 ,21 1] i ti sobserve d thati na nf-algebr a C(X)wher eX i sa compac t Hausdorff space (point- wise multiplicationo nC(X) )thi si sth ecas e indeed. Their argumentsi nfac t are sufficientt osho w that every $-algebrai n which themultiplicativ e uniti sa stron g order unit,i so fthi stype . 17.6. Theorem. Every ideal J in a Q-algebra F in which the multiplicative unit is a strong order unit, is a ring ideal. Proof:i fx e J ,yG F the n there existsa A G R such that |y|< Xe. Now |xy|= |x[|y|< |x|.Xe= X|x |e J ,henc ex yG J .Similarl yy xG J . 17.7. Remark.Th eassumptio n that themultiplicativ e unite o fF i s a strong order unito fF i snecessar yi nth m17.6 .I nth ef-algebr a C(0,1) (pointwisemultiplication )th eelemen te wit h e(t)= 1 fo ral lt G (0,1) ismultiplicativ e unit,bu tno ta stron g unit.I fx(t )= t foral l t G (0,1), thenth eidea l I isno ta rin g ideal,becaus ee = x y if y(t)= t fo r allt G (0,1). 61 InLuxembur gan dZaane n [1971,p .21 1] i ti sshow n thati nC[0,1 ] not every prime ideal isa rin g prime ideal. 17.8. Theorem. (Fremlin's theorem, Luxemburgan dZaane n [1971,p.21 1]) . In every ^-algebra C(X) where X is an arbitrary topological space (point- wise multiplication) every ring prime ideal is a prime ideal. Weno t thati nth egive n referencea proo fi sgive n under the condition thatX i scompac t Hausdorff,whic hi sno tused . Inmor e general $-algebras thm 17.8doe sno thold ,a sth efollowin g example shows. 17.9. Example. LetFb eth e$-algebr ao fal l continuous functionso n [1,°°),'whic har eeventuall y polynomials (pointwise linear operations, partial orderingan dmultiplication) ,e wit h e(t)= 1 fo ral l te [1,°° ) is themultiplicativ e unito fF . IfJ i sth elinea r subspaceo fF consistin go fal l x. €F suc h thatx is eventually polynomial with constant term equal to0 , thenJ i sa rin g prime ideal. But J is not a prime ideal, not even an ideal, because for x and y such that x(t) = 2t and y(t) = t + 1 for all te [1,°°) it holds that 0 17.10.Theorem . In every f-algebra F every ring prime ideal J is a Riesz subspace. Proof:i ti ssufficien tt osho w thatx e j implies |x|e J fo r every xe F . Ifx e J ,the nx 2e J .B yth m15.9(fc jan d (d) x2= |x2|= |x| 2, hence |x|e J ,becaus eJ i srin g prime. 62 17.11. Remark. The foregoing theorem does not hold if we take an arbitrary ring ideal. If F = C[ 0,1 ] , x(t) = t - | for all t e [0,1] and J = {xy; y£F) then J is a ring ideal; however |x| £ J, because if |x| = xy for some ye F, then y(t) = -1 for all t e [0,|] and y(t) = 1 for all t e (i,l ]. It is a rather natural question to ask for necessary and sufficient conditions for an ideal of an f-algebra to be a ring ideal. Inth eligh to fthm s 17.2,17. 3an d17. 4w eca nas k whether every ring ideal ofa nf-algebr a isnecessaril ya no-idea l ora r-idea l ora c-ideal .Th efollowin g example givesa nanswe rt othi s question. 17.12. Example.I fF i sth ef-algebr a ofbounde d real sequences with pointwise linear operations,partia l orderingan dmultiplicatio nan d Ji sth eidea l ofal l sequences converging to0 , thenJ i sa rin g ideal. Ife = (l,l,l,..), x= ( 1 ,•«f-ös •• • )e Jan de isth esequenc e such that e (m)= 1 i fm < n an de (m)= 0 ifm> n (me IN) , thenfo r f€ se q( I)wit h f(n)= e n foral ln e fj,i thold s that0 < f an dtf . Furtherw ehav ee = Lf= Lorf= Lf,bu tee £ J .Hence ,J i sno ta n o-ideal,no ra r-ideal ,no ra c-ideal . 18. Ringhomomorphismsi nArchimedea n f-algebras In this sectionw estud y connections between Riesz homomorphismsan d ring homomorphisms froma nArchimedea n f-algebra Et oa nArchimedea n f-algebra F.Furthe ra varian to fa recen t theoremo fKutateladz ei s proved, asi s avarian to fth eEllis-Phelp s theorem. 18.1. Theorem. In every Avahimedean ^-algebra F in which the multi plicative unit is a strong order unit, for a linear operator 63 we have (a) implies (b), (b ) implies (o), Co) implies (b). In the case ¥ is a ^-algebra C(X) (with pointwise linear operations, partial ordering and multiplication) and X is pseudooompaot and Haussdorff, all assertions are equivalent. Proof: (a) -+ (b): if =|((*( x+ y)) -2 (*(x)) 2- (*(y)) 2)= *(x) *(y). IfFi sa $-algebr a C(X)wit hX pseudocompac t andHausdorff , thene i sa strong order unito fF ,s oth eonl y implicationw ehav e tosho wyet ,i s Co) •+ Ca). 64 18.2. Example.Fo ra narbitrar y {-algebrawit h strong order unitth e implication (c) -*• (a)i nth eforegoin g theorem doesno thol d ingeneral . IfF i sth eArchimedea n $-algebrao fal lpiecewis e polynomialso n [1, 2] (ex.14.1 )an dee Fi ssuc h that e(t)= 1 fo ral lt e [1, 2] , thene i sa stron g order unito fF an da tth esam e timeth e multiplica tive unito fF . If $: F• +F isth elinea r operatorwhic h assignst ox e Fth econstan t termx_(0)o fth epolynomia l x (ex.14.1) , then <}>i sarin g homomorphism, but<| >i sno ta Ries z homomorphism,fo r Inth eforegoin g examplei tappear s thatth erin g homomorphism is at the same timea disjunctiv e linear functional.Th efollowin g theorem shows that thisi sa particula r caseo fa mor e general phenomenon. 18.3.Theorem . Every ring homomorphism 18.4. Example.Th e converseo fth m 18.3doe sno t hold ingeneral ,eve n ifF = K . I fE i sth eArchimedea n f-algebrao fal lcontinuou s functions xo n[0, 1] suc h that x'(0) (i.e.th erigh t differential quotiento fx inth epoin tt = 0 )exists ,the n 4>:E -* • R with cf>(x)= x'(0 )i sa dis junctive linear functional,fo rle tx e E,i f(x(t )- x(0))t " has not a fixed signfo rte [0,6)fo rever y 6> 0 ,the ni tfollow s fromth e existenceo fx'(0 )tha t x(0)= 0 an dx'(0 )= 0 .Fo r|x |simila rargu ments show that (| x| ) (0 ) =0 an d(|x|')(0 )= 0 .Thi s implies | |t(|x|)|= 11m |x|(t): |x|(0) = 1im -x(t)tx(0 ) - 65 In the second,b yth e continuity ofx ther e existsa &, e [0,6) such that x(t)> 0 o n[ O.öj) . Noww ehav e (|x|)(t)= x(t )o n[0,6^ , hence | Next,w egiv ea theore m which isa varian to fa theore mo fKutateladz e (Kutateladze [1979, thm 2.5.4 ]) an da theore mo fMeye r (Meyer [1979 , Thm3. 3 ]). The statement inth e theoremo fKutateladz e istha ti fS andT ar e linear operators froma Ries z spaceE t oa Dedekin d complete Riesz spaceF an d0 < S < T an dT a Ries z homomorphism then there exists a linear operator TT:F -* •F suc h that 0 < TT< Ip ands = TT° T NOproo f is given.A proo f for this theorem inth e caseE i sArchimedea n isgive n by Luxemburg and Schep.[1978 ,Th m4. 3]. I nthi s proof the advantagei s used that£ (E,F)i sa Dedekin d complete Riesz space. Meyer's theorem states thati fE i sa nArchimedea n Riesz spacean dFa regulato r complete Riesz space andS an dT ar e linear operators asabove ,the n there exists a linear operator TT:J -» •J suc h that0 < IT< I, andS = TT°T The proofo fthi s theorem makes useo frepresentatio n theory. 18.5. Theorem. If S and T are linear operators from a Riesz space E to an Archimedean ^-algebra F in which, the multiplicative unit & is a strong order unit, and if 0 Note that the formo fth e foregoing theorem isno t entirely the samea s the formo fKutateladze' s theorem. Inth e following chapter methods are developed which can bring this theorem inth e same form. 66 Finally,w eprov ea theore mwhic hi savarian to fth e Ellis-Phelps theorem (Schaefer[1974 ,III .prop.9. 1 ]). Firstw e have toestablis h some pre liminaries. 18.6. Definition. A subset C of a linear space L is called convex if the conditions C-,, c«e C and0 < A < 1 imply Ac,+ ( 1- A)c „£ C.c £ L is called an extremal point of C if the conditions C= Ac ,+ ( 1- A)c„ , c,,c ?E C and0 <•A <1 imply c= c ,. 18.7. Lemma, (cf.e.g . Semadeni [1971 ,lemm a 4.4.3 ]) If C is a convex set of a linear space L and C and c' are elements of C, then c is an extremal point of C if and only if c+ c '£ C together with c- c 'e C implies c'=0 . 18.8. Theorem. For a positive linear operator T from an Archimedean ^-algebra E to an Archimedean ^-algebra F such that Tep =e pj the following assertions are equivalent (a)T is an extreme point of C= {R£ £(E,F ); R >0 an d Rep= e p} (b) T is a ring homomorphism satisfying Tep =e p (c) T is a Riesz homomorphism satisfying Te.-= e,- . Proof: (a) •* (b): Thefirs t parto fthi s implicationi ssimila r toSemaden i [1971,th m4.5. 3 ],althoug h thereE = C(X )an dF = C(Y )wit hX an dY topological spaces. Lety Qb ea fixe d elemento fE ,suc h that0 < y «< e p.The n 0< TyQ< Te £= e pb yth e positivity ofT . LetS :E -* F b e the linear operator such thatS x= T(xy Q)- TxTy„ . Noww esho wtha tT +S e c an d T- S G C. (T+ S)e E= e p+ Ty Q- Ty Q= e p,similarl y (T- S)e E= e p. Further,i f0 < x e E,the n (T+ S) x= T x+ T(xy Q)- TxTy Q= Tx(e p- Ty Q)+ T(xy Q)> 0 an d (T- S) x= T x-T(xy Q)+ TxTy Q= T(x(e £- y Q))+ TxTy Q> 0 ,henc eT +S andT -s ar epositive . By lemma 18.7w e have nowS = 0 ,henc eT(xy Q)= TxTy « forever yx eE . + Now lety Qb eabounde d elemento fE ,sa y0 < y Q< Aep forsom e 0< A e R. Thenb yth eprecedin g argumentT(xA"ïy. )= TxT(A ~y n), hence T(xyQ)= TxTy Q. 67 + lety e E bearbitrary . Definef e seq(E)b yf(n )= yAne F foral ln Ne . 2 2 2 2 By lemma16. 5w ehav ey -n y+n e F> 0,s on y- ne F (a) ->- (a): LetË= { xe E;|x |< Ae £ forsom eA e F} ,F= { yeF ; |y|< ye pfo rsom ey e TR} .The nT(Ë c) Fbecause ,i fx e Ë, say |x|< Ae E forsom eA e F, the n |TX|= T|X |< ATe £= Ae.- ,s oT xFe . LetT :E -* F b edefine d byT x= T xfo ral lx e Ë,the nT i sa Ries z homomorphism. IfS i sa linea r operator fromE t oF suc h that0 < S Byth m18. 5w ehav eno wS x= TxSe F foral lx G Ë.I fT Tfro m Ft oF i s themultiplicatio n operator R then TeT Orth(F )an d§= TT°T . Supposeno wtha tT = A ^+ ( 1- A)T 2 withT ^ T2e C,A e (0,1),the n 0< AT , TT,, TT2 e Orth(F) such that AT, = TT,°T and (1 - A) T2 = TT2<>T. Ifip ,= A "TT ,an d T,eE= T 2eF= e Fw ehav e ^,eF= ^«e. .= e F,s o subset {eF>o fth eArchimedea n Riesz spaceF .B yth m12.2 1i tfollow s now that< K= i> 2>s oT ,= T 2, henceT ,an dT 2 coincideo nE . Letx e E bearbitrary . Iff e seq(E)i sdefine d byf(n )= xAn efo ral l neU, then asi nth eproo fo f (a) •*(b) w ehav e that rLf= x . r Nowb yth m11. 3w ehav e that L(T, -T 2)f =(T ,-T 2)x,henc e (T1 -T 2)x= 0 ,s o T^ =T 2. 68 18.9. Remark.Th epositivenes so fT i sessentiall y used inth eimplica tion (b) •+ (a) (compare ex. 18.2). However,i fE i ssuppose d tob e regulator complete,the n this assumptionca nb eomitted ,a sw eshal l seei nth m22.1 . 19. Normalizersan dloca l multiplierso nf-algebra s Inthi s sectionF i sa nf-algebra . 19.1. Definition. A linear operator A :F -» •F is called a normalizer on F if J forJ all xe F holds that A°£ =£ . x AX 19.2. Definition. A linear operator B :F -» •F is aalled a local multiplier on F if there exists a mapping a: F-» •F such that for xe F holds that x a(x) 19.3. Theorem. Every normalizer A on F is a Jordan operator. Proof:A = A°£ e =£^ - £(Ae)+ -£ (Ae). . Bylemm a 15.6£ (Afi)+ and £(Ae). are positive linearoperators . Ina genera l (Archimedean)f-algebr ath m19. 3i sfalse ,becaus eo na zero - algebraF ever y linear operatorA i sa normalizer ,sinc eA° £ =0 = £ for allx e F .Bu ti ti swel l known thatno tever y linearoperato ro na Riesz spacei sa Jorda n operator (cf.Meye r[1979 ,ex .2.6 .]) . 19.4. Theorem. Every orthomorphism T on an Archimedean faithful f-algebra ? is a normalizer. Proof:Le tx, ye F . First,suppos e thatT x= 0. .B yth m15. 3w ehav ex ye x .I ti sa consequenc e ofth m12.2 0tha tth enullspac e N(T)o fT i sa ban di nF ,henc ex cN(T ) and thusx ye N(T),s oT(xy )= 0 . Inthi s casew ehav e thereforeT(xy )= (Tx)y . Inth egenera l case,le ts = R° T -£ ,the ns alsoi sa northomorphism , becauses i sth e differenceo ftw o orthomorphisms. We have nowS x= < R(Tx )- (Tx) x= 0 . 69 Withth eforegoin g itfollow s thatS(xy )= (Sx)y ,henc e (T(xy))x= (Tx)xy , and sinceF i scommutativ e (T(xy))x= (Tx)yx ,henc e (T(xy)- (Tx)y) x= 0 . SinceF i sfaithfu l T(xy)- (Tx)ylx .O nth eothe r handw ehav eT xe x , hence (Tx)ys x ;x ye x ,henc eT(xy )e x .It follows that also T(xy)- (Tx) ye x ,henc eT(xy )= (Tx)y,soT° £ = £ foral lx e F . 19.5. Theorem. Every normaliser A on a faithful f-algebra ? is a stabilizer. Proof:i fxi y thenx y= 0 ,henc eA(xy )= 0 ,s o(Ax) y= 0 .Sinc eF i sfaith fulw ehav eAxly ,henc eA i sa stabilizer . 19.6.Theorem . Every orthomorphism T on an Archimedean f-algebra F is a local multiplier. Proof:Le txe F b earbitrary ,the n £° Ti sa northomorphis man d N(JC"T )3 x1. For X = {x.x1} holds that X11 = F. (V)x =(Tx) x =x(Tx ) = (VT)X- Ify e x, the n (ï^y =(Tx) y= 0 an d(£ x°T)y= x(Ty )= 0 henc e (•cTx)y= (VT)y . Itfollow sno wb yth m 12.21 thatX x°T= £ Tx> soT i sa loca l multiplier. 19.7.Theorem . Every local multiplier B on a faithful Archimedean f-algebra F which is a Jordan operator, is an orthomorphism. Proof:i ti sn orestrictio n totak eB >0 . Letx, ye F b esuc h thatxA y= 0 . There existsa mappin ga :F -> •F suc h that£ °B= £ ,, ,henc e (£°B) y= £ /„\y. Wema yassum e a(x) >0 ,fo rotherwis e replace a(x)b ya(x ) (because ((By)x)+= (a(x)y) +,s o(B(y)) x= a(x) +y). xAy= 0 an da(x ) >0 impl y xAa(x)y= 0 ,s oxAx(By )= 0 . 2 Suppose xABy# 0,then ,becaus e Fi sfaithful ,xB y^ 0 ,henc e (xBy) *0 , 2 2 2 so, sinceF i scommutative ,x (By ) =£0 ,bu tthe n certainly xB y* 0,hence , sinceF i sfaithful ,xAxB y= *0 ,contradiction . Itfollow s that xABy= 0 , soB i sa northomorphism . The following theorem follows directly from the foregoing theoremso f this section. 70 19.8. Theorem. For a Jordan operator T on a faithful Archimedean f-algebra F the following assertions are equivalent. (a) T is a normaliser (b) T is a local multiplier (a) T is an orthomorphism We remark thati nBrainer d [1962]th eclas so fal l linear operatorso n an f-algebra Fwhic har enormalize ran dloca l multipliera tth esam e time,i sstudie d under the name "the normalizero fF" . Iti sprove d thereb yelementar y means thatth enormalize ro fa nf-algebr a without non-zero lefto rrigh t annihilatorsi sa nf-algebra . Since every faith ful Archimedean f-algebra satisfies this condition,w ehav eb yth m 19.8 the following theorem. 19.9.Theorem . For every faithful Archimedean f-algebra F the space Orth(F) is an Archimedean f-algebra under the operator ordering and composition as multiplication. 71 ChapterV INVERSIONS AND SQUARE ROOTS INARCHIMEDEA N $-ALGEBRAS AnArchimedea n ^-algebra can be closed more or less under taking inverses and square roots.Fo r the caseo finversio n three closure propertycondi tions are given;eac h of them seems tob e natural.Fo r the case of square roots only one closure property condition isgiven . Asa corollar y of this theorya generalizatio n ofa theore m of Ellis[196 3] and Phelps [1964] i s given. 20. Inversions inArchimedea n ^-algebras. Inthi s sectionF i sa give nArchimedea n $-algebra,s oi n particularF is faithfulb yth m 16.6 and commutativeb yth m 15.10. 20.1.Definition ,x e F is called invertible if both the following conditions are satisfied. (a) the band B generated by X is a projection band (b) there exist ay G B such that xy= P e,where P is the projection on the band B( P exists by virtue of(a)). In that case y is called an inverse of x. 20.2. Theorem. Any xe F has at most one inverse. Proof: ify , andy 2 are inverses ofx the n by definition y,,y ~e Ban d Wl =xy 2= P xe e Bx, hencey :- y 2e Bx and x{yl -y 2)= 0 .F i s faithful,henc e the latter impliesy .- y .e B . Itfollow s thaty ,= y 9. The inverse ofx i s denotedb yx . -1 Itfollow s directly from the definition that x" isinvertibl e and (x-1)-1= x . 2- 1 20.3.Lemma . For every xe F it holds that (x) exists if and only if ~-I '. . -12 2-1 XProof exists;:B yth m 15.in that3w e hav caseeB 2c (B.x ) =( x) To prove the converse,le t0 20.5. Definition.F is oalled (a) closed under boundedinversion, abbreviated toBi j if everyX >e is invertible (b) weakly closed under inversion, abbreviated to ÜI1, if everyweak order unit is invertible (a) completely invertible, abbreviated to CI,if every element ofF is invertible. Note thatfo rever y weak order unitx e F conditio n (a)o fdefinito n 20.1i s fulfilled automatically. BIwa sdefine d byHenrikse n andJohnso n[196 1] .The y proved bymean s of representation theory that regulator completeness isa 73 sufficient condition forF t o be BÎ.Her e we givea direc t proof of this fact and an example to show that this condition isno t necessary. CI is already definedb yVulik h [1948] i nDedekin d complete Riesz spaces with respect toa give n weak order unit,an d was studiedb yRic e[ 196 8] . Their results depend on Freudenthal' sspectra l theorem (cf. e.g. Luxemburg and Zaanen [1971, Ch.6 ] )whic h holdsi nDedekin d complete Riesz spaces. 20.6. Theorem. Every regulator complete Archimedean ^-algebra F is BI. Proof: Letx e F,x > e . Define f,g e seq(F)b y f(n)= xAn e and g(n)= ( e- -x ) for all ne H. + Then0 < g(n)3< V( 'e- ^ ve ) = n— ' efo nr alln e f\.Le t for all ne «a sequence s inF b e definedb ys (m )= e+ g(n )+ .. .+ g(n ), the ns isa regulato r Cauchy sequence for all ne fi ,becaus e for alle >0 n n-1 N11 o there exists anN eft suc h that(— ) e< ^ ; ifm , >m ~ >N then n e m„+il n m,n mi« c z |sn(m1)- s n(m2)|= g(n ) +.. .+ g(n ) =g(n ) (e+ g(n )+ .. . mi-m2+l n-1 m2 n-1 n-1 mi-'112+1 + g(n)1 l )< (V e)> + (V) Oe+•• •+ e) Noww e can definea sequenc es i nFb ys(n )= r (.s. Using f(n)= n e- ng(n )fo r alln € f\i t followsb yinductio n that f n +1 ( )sn(m)= n( e- g(n)"i )hold s for allm e ]N. I f for alln e INa m+1 r sequence hpi nF i sdefine db yh (m )= g(n) (me H),then L(hp,e)=0fo r alln e ] N, hence ,i ft e seq(F) isdefine db y t(n)= n ~ s(n)fo ral l r _1 1 r r ne IN, the n we have f(n)t(n )= Lf(n)n sn= n" Ln(e- h )= L(e- h p)=e . Hence, t(n)i sth e inverse of f(n) for alln e ]N. Weshal l prove now that 2 1 t isa x -Cauch y sequencei nF . Lete > 0 . IfN e f |i ssuc h thatN > - , then for all n,m e u withm > n > N w e have |t(m)- t(n) |= t(n )- t(m ) < (t(n)- t(m))f(m )= t(n ) f(m)- e= t(n)(xAme )- e =(t(n)xAt(n)me )- e = (t(n)x- e )A (t(n) m -e) 74 20.7. Example. FR,th eArchimedea n $-algebra of all sequences ofrea l numberswit ha finit e range (with componentwise multiplication) is not regulator complete (remark 15.8), but FR is 81, even CI.Hence ,regulato r completeness is not necessary in the foregoing theorem. 20.8. Example. (a) IfF , is theDedekin d complete $-algebra of all bounded sequences of 11 real numbers (componentwise multiplication),the nx = (1 ,-*, -?, ...) isa wea k order unit in F., however x" does notexist . (b)I f F„ is theArchimedea n $-algebra of all functionsx o nF such that for all te K there existsa e> 0 such thatx i s constanto n [t,t +e ) (pointwisemultiplication) ,the n F„ is lateral complete and not Dedekind complete, (section 3). Observe that F? isCI . A $-algebra which is CI has PPP by definition,howeve r PP is notauto matically fulfilled, as the following example shows. 20.9. Example. IfF i s the Riesz subspace of the ^-algebrass o fal l sequences of real numbers with componentwise multiplication, generated by cQ0 and all elementsr (ke TL) withr = (1 ,2 ,3 , ...) , thenF ha s PPP,bu tF doe s not have PP because the band of all elements ofF wit h odd components all equal to0 , is nota projectio n band in F. Provided with the $-multiplicatio n of E,F i s itselfa $-algebra . It iseas y to see thatF i sCI . 20.10.Theorem . If Fis an Archimedean ^-algebra which is fill, then for every weakorder unit UGF there exists an f-multiplication * on the underlying Riesz space such that Fprovided with this multiplication * is a ^-algebra with multiplicative unitu . Proof: Let* b e defined byx *y = xy u for all x,y G F.The n (F,*) isa Ries z algebra by the positivity of u~ . Iffo r x,y , ze E holds thatyA z= 0 ,the n xyu Az= 0 , hence x *yA z = 0 , hence (F,*) isa n f-algebra. For all xe F w e have that x* u= xuu " =x e= x ,s ou i s themultiplicativ e unit of (F,*). 75 21. Square roots inArchimedea n ^-algebras In this section we prove that for an Archimedean $-algebra tob eclose d under taking square roots,it s regulator completeness'isasuff ie ien tcon dition.Some results can be generalized to arbitrary f-algebras,bu t here we restrict ourself toArchimedea n ^-algebras.I n this sectionF i sa given Archimedean §-algebra,s oi n particularF i s faithful byth m 16.6 and commutativeb yth m 15.10. 21.1. Definition. If for x,y e F holds that y =x then y is called a square root of X. n th roots are defined by Rice[ 196 6] , [196 8] i nDedekin d complete Riesz spaces with respect toa give n weak order unit andb y Cristescu [1976, 4.3.2 ]i nDedekin d a-complete Riesz spaces. 21.2.Theorem . Any xe F has at most one square root. 2 2 2 2 Proof: iffo ry. ,y „> 0 hold s thaty .= x an dy „= x the ny ,= y« ; since F isfaithfu l wema y apply lemma 16.3,henc ey ,= y~ . i The square root ofx i sdenote db y/ xo r x2. 21.3.Lemma . If /x and /yexis tfo rx ,y e F , then also fxy exists and /xy = /x /y. 2 2 + 22 2 Proof: ifx =w , y = z for certain w,z G F, the nx y=w z =(wz ) and wz >0 , hence /xy exists and isequa l to/x /y. 21.4.Theorem .If /x for 0 21.5. Theorem. If /x and /y exist for x, ye F then x1 y implies that /x+ y exists and /x+ y = /x + /y. Proof:b yth e foregoing theorem we have that /x e B and /ye B , henc e x y 76 o /x l /y,s o/ x /y =0 .I tfollow s that(^ x+ i^) =x+ y + 2/ x /y= x +y . Together with0 < >^ x+ / y this implies that/ x+ y exist san d/ x+ y = >/x+ v^7 . In general not every positive elemento fa nArchimedea n 21.6. Definition.F is called closed under taking square roots, abbre viated to SR if /x exists for every xe F . Similart oth ecas eo fbounde d inversion (sect. 19),a sufficien t con ditionfo ra nArchimedea n $-algebraF t ob eS Ri sregulato r completeness of F. That this condition isno tnecessar y isclea ri fw etak e FR,a s in ex. 20.7.F Ri sSR ,bu tno tregulato r complete. 21.7. Theorem. Every regulator complete Archimedean ^-algebra F is SR. Proof:Th eproo fi sb ymean so fa napproximatio n procedure,whic h is a generalizationt oregulato r complete Archimedean ^-algebraso fth ewel l known Newton method for approximating zeroso fa real -o rcomple x valued functiono nF . Fo rth eproo f that there existsa limi tw ehav et ob e more careful thani nth erea l case,wher ei ngenera l theproo fi sbase d onth eDedekin d completenesso fR . The proof that our approximating sequencei smonoton ei ssimila rt o the proofo fVisse r[193 7] (seeals o Luxemburgan dZaane n [1971,p .37 7] ). However,i nth eseque l oftha t proofa propert yo fth eordere d vector spaceo fHermitea n operatorso na Huber t spacei suse d (Luxemburgan d Zaanen [1971,Th m 53.4 ]), whichi sanalogou st oDedekin d completeness ofa Ries z spacean di sno tavailabl ei nou rcase ,s ow ehav et oprocee d differently. Firstw eprov e that /k~exist sfo r everyx e [ôe,e] , where6 e (0,1] . Letf e seq(F)b edefine d inductivelyb yf(l )= e ,f( n+ 1 ) = f(n)+ ?( *- f(n) 2)fo ral l ne M. Ifz = e - x an dg = e - f ,the n ze [0,( 1- ô) e] , g(l )= 0 an dg( n+ 1 )= \{z +g(n) 2)fo r all ne H . 11 Noww e proveb y induction thatg(n )an d g(n+ 1 )- g(n )ar epolynomial s inz wit h non-negative coefficients foral l ne IN.Fo rn = 1th estate ment istrue ,becaus eg(l )= 0 an d g(2)- g(l )= -^z . Iffo rcertai n ke M ,k > 2,hold s thatg( k -1 )an d g(k)- g( k - 1)ar epolynomial s inz wit h non-negative coefficients,the n alsog(k ) issuc h apolynomial , but then alsog(k )+ g( k -1) .No ww e have thatg( k + 1)- g(k ) =|(9(k) 2 -g( k - l)2)= |(g(k )+ g( k - l))(g(k)- g( k -1) )i sa poly nomial inz wit h non-negative coefficients. Itfollow s that0 < gan d-l-g . Inorde rt oprov e thatf isa ne-Cauch y sequence in Fw e show that 0< g< (1- <5)e .0 < g(l)< (1- <5) ebecaus e g(l)= 0 .Suppos e 0< g(k)< (1- <5)e ,the n g(k + 1)= \{z +g(k) 2)< |(( 1 -6) e+ (1- <5) 2e) < j({l - 6)e + (1- <5)e )= (1- 6)e . Nowb y inductionw e have0 < g< (1-6)e . Itfollow s thatg( n+ 1)- g(n )= |(g(n )+ g( n - l))(g(n)- g( n -1) ) < (1- ô)(g(n )- g( n -1) )fo ral l neu. Nowf(n )- f( n + 1) = g(n+ 1)- g(n )< (1- 6)(g(n )- g( n -1) )= (1- 6)(f( n - 1)- f(n) ) holdsfo ral l ne M. Furthe r 0< f(l)- f(2 )= --^( x- f(l) 2) = -2"(x- e )= ^e - jx <^e . This implies that |f(n)- f( n+ 1)| < j(l -6) ne holds foral l ne n. Lete > 0. IfN e n issuc h that^( 1 - ô)w+1 ô"1< E,the n foral l n,'m e IN such thatn > m> N holds that |f(n)- f (m )| = f (m )- f(n ) = (f(m)- f( m+ 1))+ (f(m+ 1)-f( m +2) )+ ... (f(n -1 )- f(n) ) <|(1 - ô)m e +|(1 - ô)m+1 e+ ... +j( l - ô)""1 e Suppose nowx e [0, e] arbitrary.B y theforegoin gVxVn " e existsfo r every ne H, Leth e seq(F)b edefine d by h(n)= VxV n e foral ln 78 We show thath i sa ne-Cauch y sequencei nF .Le te> 0 .Tak es e K such thatN > Ze .Fo ral ln ,me M such thatn > m > M w ehav e that V (xvm e)(xVn e) exists. Further(N " in )e (xVm e) + (xVn e) - 2V (xVm e)(xVn e) < e2e, hence VxVm_1e - VxVn_1< < (ee)2. Itfollow s from thm 16.6 that VxVme ÂxV n e< ce,s oh i s an e-Cauchy sequencei n F. Letw = r Lh.Fo ral ln e K ithold s that0 < pe x) <-e ; this implies n r 1 + ows (thm7. 1(ß) ) that rLp= 0 i fp(n )= (-i e- x ) foral l ne K. Itfoi l thatfo rqe seq(F)wit h q(n)= xVn "e hold s that rLq= x . Byth m15.1 1 (b) wehav ew 2= r Lh2 Lq= x. Finally,le txe F bearbitrary . Then /xAeexist sb yth eforegoing . Byth m20. 6w ehav e that (xVe) existsan d(xVe ) =(xve ) e < (xVe) (xVe)= e ,henc eb yth eforegoin gw ehav e thatV(xVe ) exists. •1 x -1 1 (xve) = p/(xVe) " T is invertible, so (•/(xvef )' exists and is equal toxVe . -1 •11 By lemma 20.3 also ('/(xVe) exists and it holds that • (xVe)" T121-1 = [[V(xve ) xJ j =xVe ,s o SicTëexists .I tfollow sb ylemm a 21.3 that /x = /(xAe)(xve)= /xA e /xVe exists. 22.A generalizatio no fth eEllis-Phelp s theorem The following theoremi sa varian to fth m18. 8an da generalizatio no f theEllis-Phelp s theorem (Schaefer [1974,II Ith m9. 1 ]). 22.1. Theorem. If E and F are Archimedean ^-algebras and moreover E is regulator complete then for Te £(E,F) the following assertions are equivalent 79 (a)T is an extreme point of C= {R e £(E,F);R > 0 an dRe, -= ef] (b )T is a ring homomorphism satisfying TeF= ß p (o) H is a Riesz homomorphism satisfying TeF= e F. Proof:Th e proof isessentiall y the same as the proof of thm 18.8excep t (b) •> (o) wherew ehav e to provetha tT > 0 . + 2 By thm 21.7 Ei sSR . Noww e have forever y xe E thatT x = T((/x) ) = (T(/x))>2 0 ,henc eT > 0. 22.2. Remark. Iti sobviou s that assertion (d)o fSchaefe r[1974 ,II I thm 9.1 cannot betake n intoaccoun there . 80 Chapter VI PARTIAL f-MULTIPLICATIONS INARCHIMEDEA N RIESZ SPACES In this chapter amappin g reminiscent of f-muImplication is definedo n Archimedean Riesz spaces with a strong order unit,an d a connection with orthomorphisms is demonstrated. 23. Introduction and definitions In section 15 itwa s shown that every Riesz space can be made into an f-algebra by the zero-multiplication, but itwa s remarked that for our purposes this multiplication is not useful. It is a natural question to ask whether every Archimedean Riesz space E can be provided with an f-multiplication (def. 15.4)suc h that a previously given weak order unit e of E is the unit ofmultiplication . The answer is no. IfE = c and e = (l.i.i. •••)the n Orth (E)i s Riesz Isomorphict ob (Zaanen [1975, ex.vi ]) . But b is not Riesz isomorphic to cQ,becaus e b has a strong order unit and c„ has not,henc e by thm 16.9 there exists no f-multipl icatio n on c„ such that e is the unit of multiplication. Hager and Robertson [197 7] ask for necessary and sufficient conditions for the existence of such an f-multiplication,bu t the authors remark that in general they had little success in identifying such "rings in disguise". In section 24 this question is answered forArchimedea n Riesz spaces with a given strong order unit. From Conrad [197 4] it can be deduced that in fact the ring structure of an Archimedean ^-algebra is determined by the Riesz space structure of E, because he proved that if Ei s an Archimedean $-algebrawit h f-multipli cation #,an d * is another f-multiplication on E,the n x*y=x#y#z for all x,ye Ean d some fixed ze E .Hence ,i f there exists an f-multipli cation on a given Archimedean Riesz space Ewit h awea k order unit e such that e is the multiplicative unit,the n there exists no other f-multiplica tion on Ewit h the same multiplicative unit. The foregoing is a reason to study Archimedean Riesz spaces on which is defined a partial f-multiplication or a partial ^-multiplication,o f which the definitions are given below. To avoid the situation that too many products vanish,w e require a kind of faithfulness of the partial f-multipli cation. 81 23.1. Definition (cf. Veksler [1967 ] )Apair (E,*) is called an Archimedean Riesz space with a partial f-multipliaation (abbreviated to Archimedean pf-algebra) if E is an Archimedean Riesz space and * is a mapping from ExE toEu{0} such that for all x,y,z e E holds that (we write x*y in stead of *(x,y); (PF1) x*y = y*x (PF2) if x*y * and y*z ¥= For fixed * we abbreviate (E,*) to E. * is called a pf-multipl i cation on E. 23.2. Definition. An Archimedean pf-algebra (E,*) is called multiplication complete if x*y ¥= Note that if * is a pf-multipli cation on an Archimedean Riesz space E, then it follows from (PF6) that *-1(E) 3 {(x,y) e ExE; xly}. 23.3. Theorem. On every Archimedean Riesz space E a p f-multiplication can be defined. Proof: Let * be the mapping from ExE to Eu{^} which assigns to (x,y) € ExE the element 0 if xly andtf> otherwise . Then it follows from the elementary properties of orthogonality that * is a pf-multiplication on E. 23.4. Theorem. Every multiplication complete Archimedean pf- algebra (E,*) is a faithful Archimedean f-algebra and conversely. Proof:-*-:i fx,y, ze E such thatyA z= 0 ,the ny* z= 0 ,henc e xly* z ,s o x*(y*z)= 0 . Itfollow s that (x*y)*z= 0 ,henc ex*ylz . Togetherwit h x*y> 0 an dz > 0thi s implies x*yAz= 0 .Th eothe r equality follows fromth efac t that (x*y)*z= 0 implie s (y*x)*z= 0 ,s o(y*x)lz , and thepositivenes s ofy* xan dz .(E,* )i sfaithfu l by (PF6). «-:(PF1 )follow s from thm 15.10, (PF4)follow s from thefac t thatE i sa Riesz algebra, (PF6)follow s from thefaithfulnes s ofE .Th eremainin gPF' s areeasil y verified. 82 Now we state a result similar to thm 15.3. 23.5. Theorem. If x and y are elements of an Archimedean pf-algebra E such that x*y ^ Veksler [1967 ] gives an interesting list of properties an Archimedean pf-algebra can have. We give this list here and we show that none of these properties follows from the axioms for an Archimedean 'pf-algebra. Our notation is slightly different from Veksler's. 23.6. Definition. An Archimedean pf-algebra E is said to have property A. (normality of the multiplioation) if x*y + . X,*y, =£ x*y * 0. - Pzx*Pzy # 23.7. Examples. (a) Let E be the Riesz space of all piecewise linear continuous functions x on [0.1 ] , i.e. to every x e E there exist real numbers x.(x) such that 0 = x (x) < ... < T ,(x) = 1 such that x coincides with a linear function x. on every[T-(X),x.+,(x)) (i = 0, ..., n - 1) andwit h a linear T x T x function xp on [ n( )» n+1( ) 1- Let e(t) = 1 for all t e [0,1 ]. Define x*y = z for x, y, z e E if for all t e [0,1 ] holds that x(t) y(t) = z(t); if for x, y e E there exists no z e E such that x*y = z, then define x*y = 83 butdoe sno thav e property Abecaus e e*e î 2 (b)Le tE b eth eRies z space (K ,< )wher e< isth ecomponentwis e partial ordering. Define * from ExE to Eu{0}by x*y = 0 if xiy, x*y = (JAy.Ay) if x = (2 A, A) and y = (2 y, y) ( A, ye R), x*y = ( Ay, Ay) if x = ( A, A) and y = (v,v) (^,ye IR)an dx* y= |y11 < |y|,howeverx* y= (|,0 )an d x^y^ = (1,0). 2 (c)Le tE b eth eRies z space (IR, < )wher e< isth ecomponentwis e partial ordering. Define * from ExE to Eu{tf>} by x*y = 0 if xly, x*y = (Ay;Ay) if x = (A,A) and y = (v,v) (^,y £1R), otherwise x*y = 0. Then (E,*) is an Archimedean pf-algebra which has PPP. If z = (1,0), x = y = (1,1) then x*y = (1,1) e E, V_K = Pzy = (1,0) and (P2x)*(Pzy) = (d) Let E be the Riesz space s of all sequences of real numbers. We define x*y to be the componentwise product of x and y if |.s x-y• | < °° , otherwise x*y = (e) The example (d) shows at the same time that (E,*) does not have property E. However, there exists an example of an Archimedean pf-algebra which does have property 0, but not property E. This example is given by Veksler [1967 ]. Let E be the set of all pairs [f,g ] where f and g are continuous functions from [0,1 ] to the two-point compactification K ofK, such that there exist A,y,ve]R and $,\pe C [0,1 ] (all depending on [f,g ] such that for all t e [0,1 ] , t* | holds that f(t) = 4>(t) + 7 + It - M and g(t) = *(t) + * + —^~ {t-ir it- 84 Eca nb emad e intoa nArchimedea n Riesz spaceb ycomponentwis e pointwise operations. Let[fj.g j]*[f 2,g2] b eth ecomponentwis e pointwise product (whereb ydefinitio n -°°.0= <=°. 0 =0.- »=0. «= 0 )onl yi fthi s product is againa membe ro fE ,otherwis e [f-,,g,]* [f?,g? ]= 4>. Then (E,*)i sa nArchimedea n pf-algebra. Iff :[0, 1] -*M issuc h that f(t)= , . l ,, fort* J ,f(J )= » , lI " 21 then,i fx= y = [ f,- f ],w ehav ex* y= [f 2,f2 ]e E,bu tx*|y |= (f) (Veksler [1967 , ex.2] )Le tEb eth eRies z space ofal l sequences of real numbers. Definex* yt ob eth ecomponentwis e producto fx and y onlyi fmi n( sup{x(n) ;n e N} , sup(y(n) ;n e N})< <*> ,wher e M= { ne]N ;x y ^ 0},an dx* y= o (g)Le tEb eth eRies z space (K ,<) where< i sth ecomponentwis e partial ordering. Definex* y=0 i fxly ,x* y= (-Ay ,-Ay )i fx= (A,-A ) and y= (u,-u)(A, y e]R), otherwisex* y= 0 . Then (E,*)i sa nArchimedea n pf-algebra. Ifx = (1,-1) ,y = (-1,1) , then (x+Ay+)V(x_Ay")= 0 ,henc e ((x+Ay+)V(x"Ay"))ii = {0}. However, (x*y)+ = (1,1), hence (x*y)+ii = E. This implies that (E,*) does not have property G. 23.8. Theorem. Every Archimedean pf-algebra E which is multiplication complete has properties A,BiC,D,E,F and G. Proof: A,C and E are evidently fulfilled. (E,*) is an Archimedean f-algebra by thm 23.4.Henc e (E,*)doe s have property Db yth m 15.9 (b). (E,*)doe s have property B, becausei f|x, |< |x|an d|y, |< |y|the nb y (PF4)w ehav e(|x |- IxJ)*^^ >0 an d|x|*(|y |- ly^ ) >0 ,henc eb y property!) we have Ix^yJ = lxi I *l^ i I < lxl*|yil < lxl*|y| = lx*y|- 85 (E,*)doe s havepropert y Fb yth m15. 9 (g). Itwa s provedb yVeksle r[ 1967 ,prop .4 ,prop .7 ] that properties Dan d Ftogethe r imply property G,henc e (E,*)doe s have property G. 23.9. Definition. For every x in an Archimedean pf-algebra (E,*) let V ={ ye E;x* y= £0 } and «x =( ye E;x' z# • for all ze E with \z\ <|y| } Let K(E)= { xe E;x* y* 0 /o ra H ye E};K(E ) is called the kernel of (E,*)- Fromex.23. 7 (e)i tfollow s that V isno talway sa Ries z subspaceo f E. However,w ehav eth efollowin g proposition. 23.10. Proposition. In an Archimedean pf-algebra E which has property E every V( xe E) is a Riesz sübspace of E. Proof:B y(PF3 )an d(PF5 )w ehav e thatV isa linea r subspaceo fE .I f ye V , the nx* y# 0 ,hence ,b ypropert y E,x*|y| '* 0 ,s o|y |e V . A X Itfollow s thatV isa Ries z subspaceo fE . 23.11. Proposition. In an Archimedean pf-algebra E every W (xG E) is an ideal of E. Proof:Fro mth edefinitio no fW itfollow s that |y.|< |yJ andy.e y imply y1 e WX- e Iti ssufficien tt osho wno wtha ty ,,y 2 W andX ,,X „e ] Rimpl ytha t A;.y1+ X„y 2e W . Le tze E b esuc h that |z|< l^y,+ A„yJ , then certainly hence |z|< (1^1 +i)|y x!+(l> 21 +i)|y 2l. + z < (Ï X 11 + l)|yi| +(|X 2|+ l)|y 2|. Withth m3. 4Ca ji tfollow sno wtha t z+ = z] + z\ such that0 < z]< (IX.I+ 1)|y. L hence 0< (|X.|+ l)"1 z\< |y.| (i=1,2) . Itfollow s that x*(|X,|+ l)"1 z]*4>, butthe n also x*z]_# 0 (i= 1,2) . With (PF3)w ehav eno wx* z ^0. Similarly x*z"= £0 ,henc e x*z^ 0 ; this implies that X..y..+ X„y 2e W . 23.12. Theorem.T n every Archimedean pf-algebra (E,*) which has property E the set K(E) is a Riesz sübspace of E. With the induced pf-algebra structure K(E) is a faithful Archimedean f-algebra. 86 Proof: K(E)= n{ V; ye[ ) because ifx e K(E)the n x*ye E fo ral l y e E,henc ex e v foral l ye E.Conversel y ifx e V foral ly e E thenx* y * (K(E), *K)i sa nArchimedea n pf-algebra. But (K(E), *„)i smultiplicatio n complete,hence ,b yth m23. 4w ehav e that (K(E), *K)i sa faithfu l Archimedean f-algebra. 23.13. Definition. A linear subspace J of an Archimedean pf-algebra (E,*) is called a pf-ideal if x e J, y e E and x*y t 4> imply x*y e J. InArchimedea n pf-algebraswhic h aremultiplicatio n completeth e pf-ideals are precisely therin g ideals. 23.14. Definition. A tripel (E,e,*) is called an Archimedean Riesz space with a partial ^-multiplication (abbreviated to Archimedean ^-algebra) if (E,*) is an Archimedean pf-algebra and e is a weak order unit of E, such that for all X£ E holds (P$) x*e= x . Then there is only one suchen e,which is called the p$-unit of (E,e,*). For fixede an d*w eabbreviat e (E,e,*)t o E. *i scalle d thep$-muImplicatio no fE . 23.15.Theorem . Every multiplication complete Archimedean p$-algebra (E,e,*) is an Archimedean ^-algebra and conversely. Proof:-<-:B yth m23. 4w ehav e thatE i sa nf-algebra . From (P 87 Of particular interest are the p$-multiplication s studied by Vulikh [1948 ] and Rice [ 1968 ], which are defined intrinsically on a Dedekind complete Riesz space with respect to a given weak order unit. In connection with this subject we mention Tucker [ 1972 ]. In the following section we provide another class of Riesz spaces with a p$-multiplication , namely the class of Archimedean Riesz spaces with a strong order unit. A pleasant accidental circumstance there is that property E is always fulfilled. 24. Intrinsically defined p$-multipl i cation on Archimedean Ries;z spaces with a strong order unit In this section Ei sa nArchimedea n Riesz space ande i sa fixe d strong order unito fE . LetR b eth ese to fal l standard Riesz homomorphisms $:E -*]R . By thm 12.6w ehav e thatR i snon-empty,an d infac t therei sa1- 1 correspondence betweenR an dth ese tJ o fal lmaxima l idealso fE , and for thelas t seti thold s that nj= {0} .Hence ,i tfollow s from (fix=0 for all<| > eR tha tx = 0 . We definea mappin g :Ex E-» •Eu{<£ }b y*(x,y )= z fo rx,y, ze Ei ffo ral l <|>e R hold s that 24.1.Theorem . If E is an Archimedean Riesz space with a strong order unit &, and * is the mapping as defined above, then (E,e,*) is an Archimedean p$-algebra. Proof: (PF1) if x*y = z for x,y,z e E, then for all 4>eR we have that «f>(x)c|>(y) = 88 If x*(y*z)= (PF5)i fX ¥=0 an d x*y* 4>, then for all 24.2. Definition. The mapping *, defined above is called the intrinsically defined p9-multiplication on E (relative to e). In the sequel the symbol *i sreserve d for this intrinsically defined p$-multiplicationo n E. Noww eshal l examine which properties ofVeksler' s list (def. 23.6) are fulfilled inth e caseo fou r intrinsically defined p$-multiplication. 24.3. Proposition. Property A is not fulfilled in general. Proof:W etak e the Riesz space Eo fex . 23.7(a). Asa consequenc eo fth e compactness of[0, 1 ] the only realvalued Riesz homomorphisms $ with 4>(e)= 1 ar e thepoint-evaluation s<|> t(te [0,1 ]) , i.e. (f>t(x)= x(t ) forx e E(cf . e.g. Schaefer [1974 , III.1 Ex .1 ] ) . Hence,* i sexactl y the pf-multiplicatio n as defined inex . 23.7(a) , hence also (E,*) does notsatisf y property A. 24.4. Proposition. Property B is always fulfilled. Proof: Let •Klxj'yjl) = Mx^y^l = |4>(x1)d>(y1)| = | 89 24.5.Proposition . Property C is always fulfilled. Proof: Suppose x,y,ze Esuc h that x*y + By elementary projection propertiesw ehav ee A( e- e ) = 0 henc e e z*( vz e -e ')= 0 ,an' d I- P =p e_ e-e_ Furtherw ehav ee * e= e . With (PF3)i tfollow s thate * e =e , henc e foral l §e Rw ehav e We have <()(P x) = c|>(P x) = c(>(sup{xAne ;n e IN}) > sup{<|>(xAne );n e IN} = sup{ 0 if 4>(ez)'= 0 *(ezH(x) ,.. (u) (x) if Noww eshal l prove thati n(11 )i nfac t equality holds. Wed otha tb yobservin g that *((I- P z)x)= *(( I- P e )x)= «>(P e_ex ) = ())(sup{xAn(e- e ); neM} )> sup{ .•(x) if «t>(e - e7) = 1 >(x) if 4>(e ) = 0 (n)an d(im )giv eno wtha t <|>(x)= *(P x +( I- P z)x)= 90 Nowi tfollow s that e_,*x= P zx. Similarlyca nb eprove d thate * y=P y an de *(x*y )= P (x*y). Finally, Pz(x*y)= e z*(x*y)= (e z*ez)*(x*y)= e z*(ez*(x*y))= = ez*((ez*x)*y)= e z*(y*(ez*x))= (e z*y)*(ez*x)= (e z*x)*(ez*y)= P zx*Pzy. 24.6.Proposition . Property D is always fulfilled. Proof: Ifx* yG Ethe n foral l 4,G Rw ehav e 4>(|x*y|)= 14>(x*y>| = l 24.7. Proposition. Property E is always fulfilled. Proof:Le tx, ye Eb esuc h that x*ye E.e i sa stron g order unito f E, hence there existsa A G M such that |x|< Ae,s o- x< Ae,s ox + A e>0 . Ae*ye Eb y(P$ )an d (PF5).S ob y(PF3)( x+ Ae)* y eE . The foregoing proposition nowgive s that |x+ Ae|*|y |e E,s o( x+ Ae)*|y |e E . (-Ae)*|y|e E;i tfollow s by(PF3 ) that also x*|y| eE . 24.8.Proposition . Property F is always fulfilled. Proof:Fo ral l cf> e Ri s 24.9.Proposition . Property G is always fulfilled. Proof: InVeksle r [1967, prop.4 ,prop .7 ] i ti sprove d that properties D and Ftogethe r imply property G,henc e (E,*) does have property G. 24.10.Theorem . If E is an Archimedean Riesz space with a strong order unit e., and E is moreover a ^-algebra such that e is the multiplicative unit of Ej then the intrinsically defined ^-multiplication * on E (relative to e) coincides with the ^-multiplication, hence in particular (E,*)is multiplica tion complete. Proof: Let #b eth e$-multipl icatio no nE * Ifx,y, z G Ear esuc h that z= x # y ,the nb yth m 18.1w ehav e foral l 24.11.Theorem . An Archimedean Riesz space E with a given strong order unit e can be provided with an f-multiplication such that e is the multiplicative unit if and only if (E,*) is multiplication complete, where * is the intrinsically defined pi-multiplication on E (relative to e). 91 Proof: if E is a $-algebra with multiplicative unit e, then by thm 24.10 multiplication * coincides with the 25. Orth(E) forE a nArchimedea n Riesz space witha stron g order unit In this section,w eprove , independently of Bernau [1979 ],that Orth(E) isa nArchimedea n f-algebrai fE i sa nArchimedea n Riesz space witha strong order unite . Let* b eth eintrinsicall y defined p$-multiplication onE (relativ et oe ) (section 24);le tK = K(E )b eth e kernel of(E,*)(sectio n 23),the nb y thm 23.12w ehav e thatK wit hth einduce d ordering and multiplicationi s an Archimedean f-algebra. LetZ = Z(E )b eth e classo fal l centre operatorso nE (sectio n 12),then provided with the operator orderingan d composition asmultiplication ,Z i sa partiall y ordered algebra. 25.1. Theorem. For every x£ K the mapping £ from Et oE which assigns to y£ E the element x* y of E, is an element of Z; conversely, for every Te Z there exists anx e K such that T= £ x Proof: •+: i fx e K,the no naccoun to fth e fact thate i sa stron g order unito fE there existsa A e F such that |x|< Ae. For ally € E+w e have nowb ypro p 24.4 that |£y| = |x*y|<|Ae*y|=Ay, hence £ is A X a centre operator. Conversely, ifT e z, then there existsa A e ]Rsuc h that |Tx|< A xfo r all xe E. W eshal l prove now thatT = £ . Therefore,w ehav et osho w thatT ee K.I fx e Ean d<( >e R are arbitrary and ify = x - 25.2. Theorem. The linear operator A from K to Z which assigns to xe K the element X of Z is an order isomorphism. Proof:A i sinjective ,becaus eX =X forx ,y e Kimplie sX e = Xe, hencex = y .A i ssurjectiv eb yth m 25.1.Ai s positive becausex > 0 impliesx >0 ,A ~ ispositiv e becauseX >0 implie sX e > 0,henc e x >0 .I tfollow s thatT i sa norde r isomorphism. 25.3. Theorem. For every Archimedean Riesz space E with strong order unit e the space Orth(E) is an Archimedean f-algebra under the operator ordering and composition as multiplication. Proof:b yth m12.2 3w ehav e Orth(E)= Z(E) .K i sa nArchimedea n Riesz space byth m23.12 , hence,b yth m25. 2 Z(E)i sals oa nArchimedea n Rieszspace . A isno tonl ya norde r isomorphism,bu tals oa nalgebr a isomorphism, because,i fx ,y e K,the nA( x* y )= £^y =X x °X y = (Ax)(Ay). Itfollow s thatZ i sa nArchimedea n f-algebra. 93 REFERENCES Abramovic',Ju.A. ,Veksler , A.I.an dKoldunov , A.V. [1979] On operators preserving disjointness, (Russian), Doklady Akademii Nauk SSSR, Tom. 24.8,No .5 ,pp . 1033-1037. English transi.: Soviet Math. 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[1967] Introduction to the theory of partially ordered spaces, English translation from the Russianb yL.F . Boron,Groningen.(Wolters - Noordhoff Scientific Publ. Ltd.),x v+ •38 7pp . Wickstead, A.W. [1979] Extensions of orthomorphisms, preprint, The Queen's University ofBelfast ,2 0 pp. Zaanen,A.C . (also Luxemburg and-) [197 5] Examples of orthomorphismss J. Approximation Theory, Vol. 13,pp . 192-204. 99 SAMENVATTING Indi t proefschrift worden lineaire operatoren op Riesz ruimten bestu deerd,i n het bijzonder disjunctieve lineaire operatoren. Enkele voor beelden van niet orde begrensde disjunctieve lineaire operatoren van een Riesz ruimte op zichzelfworde n gegeven,terwij l de orde begrensde disjunctieve lineaire operatoren in verband worden gebracht met een par tieel gedefinieerde vermenigvuldigingsoperatie in de Riesz ruimte. Hoofdstuk Igeef t een inleiding in de theorie van Riesz ruimten. In hoofdstuk IIworde n enkele typen van convergentie bestudeerd inver band met lineaire operatoren. Disjunctieve lineaire operatoren worden in hoofdstuk III bestudeerd. In het bijzonder wordt aandacht geschon ken aan disjunctieve lineaire functionalen en orde begrensde disjunc tieve lineaire operatoren van een Archimedische Riesz ruimte met een sterke orde eenheid op zichzelf. In hoofdstuk IVworde n Riesz ruimten bestudeerd die voorzien zijn van een vrij star gedefinieerde vermenig vuldigingsstructuur, de zogenaamde f-algebra's. Enige overeenkomsten en verschillen tussen Riesz homomorfismen en ring homomorfismen worden aangegeven en een variant van de stelling van Ellis-Phelpsword t afger leid. Met behulp van de in hoofdstuk V ontwikkelde theorie van inver- teren enworte ltrekke n in regulator complete ^-algebra's wordt een generalisatie verkregen van bovengenoemde stelling. In hoofdstuk VI worden partiële vermenigvuldigingsoperaties opArchimedisch e Riesz ruimten met een zwakke orde eenheid axiomatisch ingevoerd. Op iedere Archimedische Riesz ruimte met sterke orde eenheid wordt tenslotte een intrinsieke partiële vermenigvuldiging aangegeven en in verband ge bracht met de ruimte van alle orde begrensde disjunctieve lineaire ope ratoren van de Riesz ruimte op zichzelf. 100 CURRICULUM VITAE Brand van Putten werd geboren op 15jun i 1954 te Zwolle. Mei 1971 legde hij het HBS-B examen af aan het Ichthus College te Enschede en werd in datzelfde jaar ingeschreven als student aan de Rijksuniversiteit te Utrecht. Juli 1974 legde hij het kandidaats examen wiskunde met bijvak natuurkunde af. Als specialisatie koos hij de numerieke wiskunde en werd daarin onderwezen door Professor Dr.A . van der Sluis. Juni 1976 legde hij het doctoraal examen af met uitgebreid hoofdvak wiskunde en klein bijvak muziektheorie. In oktober 1976 werd hij benoemd totwetenschappelij k ambtenaar aan de Landbouwhogeschool teWageningen ,secti e zuivere en toegepaste wiskunde. Per 15septembe r 1977wer d hij in staat gesteld om als wetenschap pelijk assistent het onderzoek te verrichten,da t tot dit proefschrift heeft geleid. Sinds 1septembe r 1980 is hij alswetenschappelij k ambte naar verbonden aan de Landbouwhogeschool te Wageningen, sectie statis tiek. 101 ERRATA toDisjunctiv e linear operators andpartia l multiplicationsi n Riesz spaces -B .va nPutten . 46 :x, ye E+ should read x,y> 0 4 ' ' :sup(x.-x )shoul d read sup{x,-x} 8~ . : (f) a polar P of E is called a principal polar if there exists a Z e E such that P = z11 92,3 : should be deleted 9 ' : The polar z is called the principal polar generated by z, and is denoted by P . 11 : Def. 3.14 should read The intersection of all bands which contain an element z of a Riesz space E (which is a band by thm 7>.?) is called the principal band generated by Z. Every band of this form is called a principal band. E is said to have the projection property (abbreviated to PP,I if every band of E is a projection band. E is said to have the principal projection property (abbreviated to PPPJ if every principal band of E is a projection band. 1312 : 5 should read M oo Z + 16 : = ( T (Xe + I X.ei - XeN))M >0 i*N 16, :T xshoul d read fr x) 3 28 :a Ries z space should read anArchimedea n Riesz space 28 :y e rLf should ready e cLf 35- :T < pT< |p|lshoul d read T< ui< |u|l 3715 :I nthi s section 14shoul d read Insectio n1 4 11 11 383 :X =X shoul d read X =E 40.,,, :shoul d read Inex.13.1 3th eRies z subspace Ti sprime . Note that T isno ta nideal . 413 : A j V x J afeJ1 iel CT(1,K1 keK 4315 : NO Jj then N=J should read N D M, then N=M 5 44 thm 6.12 should read thm13.1 2 44 + 14 x e E should read x e E 44, I should read I 5 11 45 y A(e-y) = 0 should read y A(e-y) eR 46! n s DR should read n z DR 47 11 By the foregoing theorem should read By thm 13.22 6 51' for all x e E should read for some x e E 6 52' T(f(n))(0) should read T(f(n))(l) n themaximu m should read n greater then themaximu m 9 2 2 2 2 58 13 xy - ny x+ n y shoul d ready x -ny x+ ny 64 3 y should read $ 66 16 S = TT* T if J = Id(T(E)) 66 13 Tzsx = TxSz holds for all x,z e E. 66 10-1 e should read z 69 12 on F should read on a O-algebra F 72 (for a definition of P see Luxemburg and Zaanen [1971, thm24. 5 ]). 78 < i(l-5)ne should read < }(l-6)n_1e 78 13 i(l-6)N+1 5"1 should read £(1-">)V1 78g on < J(l-6)m i (l-S)Ke < J(1-Ä)K(S e < ce 7911 k=0 thm 7.1fe; should readth m 7.1(d) 8317,18 C. if B projection band( ze E),x* y *0 - *P x *P y *0 8515 min (sup{|x(n)|;n e W}, sup{|y(n)|;n eN} ) 90J + Pts = sup {s An|t|; n € W} - sup {s"An|t|; n e W} 90 16 We have should read Assume first x > 0, then we have 911 Now it follows that e *x = P x for all x > 0, hence for all x GE we have P x = P x - P x e *x e *x = e *x. 92. : 0 < {xy), thenc > 0 . Hence <|>((ce- x)( e+ y) )= 0 .No ww ehav e 0 < (ce- x) + <;(c e- x) +(e+ y )= ((c e- x)( e+ y) ) e J,whic h implies (ce- x) + e J,henc e a narbitrar y Riesz homomorphism from Ft o1 R with , y < x y xx*y: * 0 - lv i' l * l- C. if for certain z e E holds that the band B is a projection band and