Nonstandard methods in geometric functional analysis

Amer. Math. Soc. Transl. (2) vol. r5r, 1992 NonstandardMethods in GeometricFunctional Analysis UDC517.11-517.98 A. G. KUSRAEVAND S. S. KUTATELADZE Nonstandard methods in the modern senseconsist of the explicit or implicit appeal to two different models of set theory-"standard" and "nonstandard"-1s investigateconcrete mathematical objects and problems. The main development of such methods dates to the last thirty years, and they have now crystallized in several directions (see 1291,l42l and the bibliography cited there). The main directions are now known as infinitesimal and Boolean analysis. In this paper we shall outline new applications of nonstandard methods to problems arising in the area of our personal interests, grouped together under the generalheading of geometric functional analysis [48]; we shall also point out some promising directions of further research. $1. Infinitesimal analysis 1.1. Infinitesimal analysis,following its creator A. Robinson, is frequently referred to by the expressivebut rather unfortunate phrase "nonstandard analysis"; nowadays one most frequently speaksof classicalor Robinsonian nonstandardanalysis. Infinitesimal analysisis characterizedby the use of certain conceptions,long familiar in the practice of natural sciencesbut frowned upon in twentieth-century mathematics, involving the notions of actual infinitely large and infinitely small quantities. 1.2. Modern expositions of nonstandard analysis rely on formulas of E. Nelson's internal set theory IST [58] and its later developments,the external settheoriesof K. Hrbacek (EXT) [49] and T. Kawai (NST) t53]. From the standpoint of the "working mathematician-Philistine," the essenceof these theories is as follows. l99L Mathematics Subject Classffication. Pimary 46520, 03H05. @ 1992 American Mathematical Society 0065-9290192$1.00 + $.25 per page 9l 92 A. G. KUSRAEVANDS. S. KUTATELADZE Ordinary mathematical objects and properties are called internal (and considered, if a rigorous formalization is desired, within the framework of Zermelo-Fraenkelset theory ZFC). One introducesa new predicate St(x) , expressingthe property of an object x to be standard (qualitatively speakingobtained through existenceand uniquenesstheorems, i.e., the set of natural numbers is standard, but the infinitely large natural numbers are nonstandard). Mathematical formulas and conceptsin whose construction the new predicate is used will be called external. "Cantorian" sets endowed with external properties are referred to as external. In Nelson's theory, such sets are consideredonly as terms of a metalanguage,which is used only for convenience. In EXT and NST one can treat them as objects of Zermelo theory, which requires elaboration of a formalism and introduction of a new primary predicate Int(x) , stating that the object x is internal. The available formalisms ensurethat the extension of ZFC is conservative,i.e., when proving mathematical statementswhoseformulations do not involve external concepts,we may legitimately invoke the theories IST, EXT, and NST, as no less reliable than ZFC. 1.3. A point of crucial importance is that the new theories contain additional rules, easily motivated at the intuitive level, which are known as the principles of nonstandard analysis. We present their rigorous formulations in IST. ( 1) Transfer principle: x n )- $ x ) p ( x , x t , . . . , x r ) ) , where p is an internal formula and rp- q(x , xr, ... , xn) (i.e., p doesnot contain any free variables other than those listed). (2) Idealization principle: (v"",)"'(v"xr)((v"tr)q(x, xr ( v x r ) . . . l v x r ) ( v " o n r ) ( : x ) ( v ey z ) g @ , ! , x t , . . . , x n ) * ' ( 3 x ) ( V " t y ) e ( *! ,, x 1 , . . . , x n ) , where p is an internal formula and g -- p(x , ! , xr, ... , xr) . (3) Standardizationprinciple: ( v x r )" ' ( v x , ) ( v ' t x ) ( 3 ' t y ) ( Y z )" zt e ! * z e x A q ( 2 , x t , . . . , x n ) , where e - e(z , xr, ... , xn) is an arbitrary formula. The index st indicates that the quantifier in question is relativized to standard sets;the index st fin has the analogousmeaning with regard to standard finite sets. $2. Boolean-valuedanalysis 2.1. Boolean-valuedanalysisis characterizedby the extensiveuse of the terms lowering and lifting, cyclic hulls and mixing. The development of this trend, which emergedunder the impetus of P. J. Cohen's remarkable work on the continuum hypothesis,leadsto essentiallynew ideas and results, first and foremost, in the theory of Kantorovich spacesand von Neumann NONSTANDARD METHODS IN GEOMETRIC FUNCTIONAL ANALYSIS 93 algebras. The modeling device offered by Boolean-valued analysis makes it possible, in particular, to consider the elements of functional classesas numbers, which substantially facilitates the analysis and creates a unique possibility of automatically extending the scopeof classicaltheorems. 2.2. The construction of a Boolean-valuedmodel begins with a complete Boolean algebra .8. For every ordinal a e On one defines Vjt) :- {x:(rB e a)x:dom(x) ---+ B n dom(x) e V}B)}. After this recursive definition, one introduces the Boolean-valueduniverse V@) or classof ^B-sets: ._ v:r,. v@) u aeO, 2.3. Taking an arbitrary formula of ZFC and interpreting the connectives and quantifiers in the natural way in the Boolean algebra B, one definesits truth value [,p\, which depends on the way in which A is built up from atomicformulasx:! and x e y. Thetruthvaluesofthelatteraredefined for x, y e V@) by a recursionschema: [x e yn:- y Q ) n f i z: x \ , V z€dom(y) llx-yn:- V xQ)+[.2€yn^ z€dom(x) A yQ)+fizex\ z€dom(y) (the sign + symbolizesimplication in B) . The universe V@) with the above valuation rule is a ("nonstandard") model of set theory in the following sense. 2.4. Transfer principle. For any theorem g of ZFC, the formula [p] - 1 is valid, i.e., A is true inside V@). 2.5. In the class V@) there is a natural equivalencex - !:- [x - ln : I , which preservestruth values. In this connection, one can use a special device to go over to a separateduniverse 7@) , in which x : ! ++ l[x yn : 1 . In fact, the identification V@) :- 7@) is usually assumedwithout specialmention. The basic properties of tr'(B) are expressedby the following assertions. 2.6. Mixing principle. Let (b)er= be a partition of unity in .B, i.e., I # 4 - b e A b r : 0 , V c r = b q - 1 . F o r a n y f a m i t y ( " r ) a . = o f t h e u n i v e r s eV @ ) there existsa (unique) mixture of (xa)a6, with probabilities (b6)aE=,i.e., an element x of the separateduniverse,denotedbt Dr.= brx, or sixrrrbrxl , such that [x - xrn 2 b, for ( e E . 2.7. Maximum principle. For every formula e of ZFC there is an element xo e V@) for which : ne(xs)n. W=x)q(x)n 94 A. G. KUSRAEVANDS. S. KUTATELADZE In particular, V@) containsan object ,9 whichplays the role of the field R inside V@). 2.8. Besidesthe aboveprinciples, there is an important procedurefor passing to V@) from the ordinary von Neumann universe V , where the latter is defined by the recursion schema Voi: {x: (38 e a)x € P(Vp)}, V :- U Vo. a€On This procedure is defined by the rule on :- a, im(xn):- {l}. d o m ( x n )r - { / n r y e x } , The element x^ € V@) is known as the standard name of x . We thus have a canonical embeddingof V into V@) . Apart from this we have a technique of lowerings and liftings of sets and correspondences. 2.9. Given an element x e V@), its lowering x I is defined by the rule x | :- {t e V@l:[/ e xn : l]. The set x J is cyclic,i.e., closedwith respect to mixing of its elements. 2.10. Let F be a correspondencefrom X to I inside V@) . There exists a correspondencef' | -and it is unique- from X I to I I such that for any subsetA of X inside V@),we have F(A) J : F I (1 l) . In particular, a map f : Xn + Y inside V@) defines a function-lowering f I : X - + Y I s u c h t h a tf J ( x ) - f ( x " ) ( x e X ) . 2 . l l . L e t x € P ( V @ ) ) . D e f i n eA l : - A a n dd o m ( x t ) - x , im(xJ) The is element x f called the lifting of x. It is easyto seethat x 1l {1}. is the least cyclic set containing x,i.e., its cyclic hull: x lJ : mix(x) . 2 . l 2 . L e t X , Y e P 1 V @ l )a n d l e t F b e a c o r r e s p o n d e n c e f o r m X to Y. There exists a correspondence^F | -and it is unique- from X I to Y I inside V@) such that dom(,F 1) : dom(f') f and for every subset A of dom(f') we have F I @ I) : F (A) 1 if and only if F is extensional,i.e., ! 1 e F ( x , )r l l x r: x . r n3 V W1: !2n. YeF@r) Inparticular, amap f:X ---+YJ generatesafunctionf I:X" --+Y such that f t(x") -f(x) for x e X.If necessaryinspecificcases,thelowering and lifting procedurescan be iterated. $3. Vector lattices 3.1. There are severalexcellentmonographson the theory of vector lattices [4], [18], [19], [55], [70]. Vector latticesare also commonly known as Riesz spaces,and order-completevector lattices as Kantorovich spacesor K-spaces. A K-space is said to be extended if any set of pairwise disjoint positive NONSTANDARD METHODS IN GEOMETRIC FUNCTIONAL ANALYSIS 95 elementsin it has a supremum. The most important examplesof extended K-spacesare the following: ( 1) the space M (O , 2 , lt) of equivalenceclassesof measurablefunctions, where (O, I, 1r) is a measurespacewith lt a o-finite measure(or, more generally,a spacewith the direct sum property, see[18]); (2) the space C*(Q) of continuous functions defined on an extremally disconnectedcompact space O with values in the extendedreal line, taking the values *oo only on a nowheredenseset [4], [9], [55]; (3) the space V of selfadjoint (not necessarilybounded) operatorsassociated with a von Neumann algebra(see[66]). To savespace,we shall restrict attention to the real case,since the analysis of complex K-spacesis entirely analogous.The symbol P(E) will denote the Boolean algebraof order projections in a K-space E . If E contains an order unit, C(E) is the Boolean algebraof unit elements(fragmentsof the identity) in E. The algebrasf@) and €(,8) are isomorphic and known as the base of E. Throughout the sequel, B will be a fixed complete Boolean algebra. The basisfor Boolean-valuedanalyris of vector lattices is the following result. 3.2. Theorem (Gordon [6]). Let ,9 be the field of real numbers in the model V@) . The algebraic system .q I Q.e., .q with lowered operations and order) is an extended K-space. Moreover, there exists an isomorphism X of the Boolean algebra B onto the base ry@) such that b < [x : Y\,- x(b)x - x(b)! , b < fix < yn *- x(b)x < x(b)y forall x,ye .q I and be B. Throughout the sequel, R will denote the field of real numbers inside V@). If the base of a K-space E is isomorphic to .8, then E itself is isomorphicto the foundation Eo C.q l, and in this situation ^E is extended if and only if E0 - SE J . Under thesecircumstancesone saysthat ,% J is a maximal extension and I a Boolean-valuedrealization of the K-space .E . It is noteworthy that Boolean-valuedrealizationsof certain structureslead to subsystemsof the field I . 3.3. Theorem[25]. (l) A subgroupof the additive group of ,q is a Boolean-valuedrealization of an archimedean lattice-orderedgroup. (2) A vectorsublatticeof ,q , consideredas a vectorlattice over thefield R^ is a Boolean-valuedrealization of an archimedeanvectorlattice. (3) An archimedean f-ring contains two mutually complementarycomponents,one of which is a group with zero multiplication realized as in (l), and the other has a subring of the ring ,9 as a Boolean-valuedrealization. (Q An archimedean f -algebracontains two mutually complementarycomponents,one of which is a vectorlattice with zero multiplication realizedas in 96 A. G. KUSRAEVANDS. S. KUTATELADZE (2), and the other is realized as a subring and sublatticeof ,q , consideredas an f-algebra over R^ . 3.4. Gordon's theorem implies the main structural properties of K-spaces. We shall dwell on the realization of K-spacesand functional calculus. Let Q be a Stoneancompact subspaceof the Boolean algebra B and define C*(Q) as in 3.1 (2). We call a map e:R -- B a resolutionof unity in B if (l) e(s)< e(t) (s i r) ; (2) V,.^ e(t) - 1, A,.^ e(t) - 0 ; (3) V,.,e(s) - e(t) (l e R). Let B(R) bethesetof allresolutionsofunityin ^8. ThesetsC."(Q) and ^B(R) can be endowed canonically with the structure of an extended Kspace(see[4] and [19]). 3.5. Theorem (1291,t50l). The extended K-space g I is (algebraically and order) isomorphic to each of the K-spaces B(R) and C*(q . Under this isomorphism an element x €. g I is mapped onto a resolution of unity t - el (l e R) and onto afunction 7:Q +R by theformulas el :- [x < l"n (r e R), 7(q):- inf{/ e R: [x < /nne q] (a e 91. 3.6. Let gR and ,% (R) be the o-algebraof Borel sets and the vector lattice of Borel functions, respectively,on the real line. We identify B with the algebra of fragments of the identity rn ,9 J (see 3.2). For every x € .q I there exists a unique spectra measure (- sequentially o-continuous Boolean homomorphism F:9n - B) such that tt(-x , t) : el (l e n; . The measure p definesan integral I -*(f ) :- [ f Al drt(t) JR (f e ,q 8)). In this situation lr(f ) is the unique element of !t I for which [ . l ' , ( f )< r n n : p ( f < / ) ) . 3.7. Theorem (1291,[50]). The map I*:,9(R) + .9/ ! is the unique sequentially o-continuous lattice and algebraic homomorphism satisfying the condition /"(id^) - x. 3.8. For other aspectsof Boolean-valuedanalysisof vector lattices, see[7], [8],[24],1291, [50],[51],t651. $4. Positive operators 4.1. General information about positive and order-boundedoperatorsmay be found in 1241,1291.Take arbitrary K-spacesZ and E. A positive operator O: Z - ^E will be called a Maharam operator if it is order continuous a n d O ( [ 0, z l ) : [ 0 , O ( z ) ] f o r e v e r yz e Z - , w h e r el a , b l ; - { c :a 1 c < b } is an order interval. Let mZ be a maximal extensionof Z and D(O)* the NONSTANDARD METHODS IN GEOMETRIC FUNCTIONAL ANALYSIS 97 set of all O 1 z e mZ such that {Qz': z' e Z, 0 ( t' 1 r} is bounded. Then D(O) :- D(O)+ - D(O)* is a foundation in mZ and Q extendsto a Maharam operator on the whole of D(O) . We say that O is essentially positiveif O > 0 and O(lrl) - 0 implies z - 0. 4.2. Theorem t221. Let Q be an essentiallypositive Maharam operator. There exist a K-space Z and an essentiallypositive o-continuousfunctional e: Z + 9 in the model V@), and there existsan isomorphism t from D(O) onto the K-spaceZ I suchthat Q: e I o t. 4.3. The above result reducesthe investigation of Maharam operators to analysis of the classof o-continuous positive functionals. What is the situation with regard to arbitrary positive operators? Various approachesbased on Boolean-valuedanalysis may be adopted here. Let us consider an orderbounded operator from a vector lattice Z into E:9 l. There exists an order-bounded R^-linear functional e: Zn -f R inside V@) for which Q: g l o /, where j:z -- z^ (z e Z). The map Q -- rp is anjsomorphism of the spaceof all order-boundedoperators L,(2, ^E) onto Z L where 7 it the space of order-bounded function als on Z . In particular, O > 0 if and only if np > 0n - I . The disadvantageof this device is that the map O + p does not preserveorder-continuity. On the other hand, for a positive operator Q: Z + E one can construct an essentially positive Maharam operator O and a lattice homomorphism h:Z - D(iD) suchthat Q: @o h,where the pair (h,6) is minimal in a certain sense(see[1]). Appealingto Theorem 4.2, we obtain a representation Q : e I o r ' , w h e r e t ' : - h o t a n d g i s a n e s s e n t i a l l y p o s i t i voe- c o n t i n u o u s functional in the model V@) . The disadvantageof this approach is that the space D(lD-) may prove to be invisible. However, in a fairly generalsituation, D(lD) is realized as the spaceof functions (in two variables)on P * Q, where P and O are Stonean compact subspacesof Z and ,8, respectively (see t55l). 4.4. The above argumentsare easily applied to the algebraof fragments of an arbitrary positive operator O acting from a vector lattice Z to a K-space E with filter of units 6 and base f@) (see[] and t39l). We dwell on the representation of the projection S of an operator Z onto the component g in {Afo generatedby the operator O. Let us call a set of operutors L,(Z , E) a generatingset if Ox- - sup{p@x:p € g} for all x e Z. To study interesting fragmentsby lifting into a Boolean-valueduniverse,one can reduceeverything to the caseof functionals. For the latter, using infinitesimal representations,one readily proves that Sx- inf* {" pTx; pdQx ! o , p e ,q} , Sx-inf {"Ty,O(x- y)=0, 0<y (x}, * o where is the standardization symbol, the "standard part" operation, r 98 A. G. KUSRAEVANDS. S. KUTATELADZE denotesinfinite smallnessand -.. denotesthe exactnessof the formula, i.e., the attainability of equality. Interpreting the above nonstandard representationsand performing the lowering, one arrives at the following formulas l29l: S x - s u p i n f { z T r * n oT r , 0 < y I x , n e g ( E ) , nil(x - y) < e}, e€€ S x : s u p i n f { ( T E p )Tdr : p D r 1 e , p e . q , ft q_f(E')}. e€( $5. Banach-Kantorovich spaces 5.1. A Banach-Kantorovich space consists of a (real or complex) vector spaceX,d K-spaceE,andavectornonn l.l:X + E suchthatthefollowing conditionshold: (1) the nonn is decomposable, i.e., if lxl - et+e2, where x e .X a n d e r , € z e , E + ,t h e n x : x r * x z a n d l x l r l- e o ( k : - 1 , 2 ) f o r suitable x, x2 e X; (2) X is o-complete,i.e., for any net (x") c X , if o-limlxo- *pl:0, t h e n o - l i m l x o - x l : 0 f o r s o m ex e X . W e s h a l l assumethat {lxl:x e X}od - E c.q I. lf E is extended,i.e., E :,9 L then X is also said to be extended. An example of an extended BanachKantorovich spaceis the space M(O,Z,lt,Y) of (equivalenceclassesof) strongly measurablevector-valued functions with values in a Banach space Y. 5.2. Theorem 1231.Let x be a Banach space in the model V@). Then the lowering x ! is an extendedBanach-Kantorovich space. Conversely,if X is an extendedBanach-Kantorovichspace,there exists a unique (up to linear isometry) Banach space x in V@) whoselowering is linearly isometric to X . 5.3. Let us call the bounded part of the space x I the restricted descentof x. The restricted descentsof Banach spacesin V@) constitute the classof .B-cyclicBanach spaces.Let B be the complete Boolean algebraof norm one projections in a Banach space X. We shall say that X is cyclic with respect to B, or .B-cyclic,if, for an arbitrary partition of unity (nr)rr= c B and any bounded family ("r)r.- c X there exists a unique element x e X such that nexc: ftex (( e E) and llxll < trp(.= 11x6ll . Let A(B) denotean arbitrary commutative AW* -algebrawhose complete Boolean algebraof idempotents is isomorphicto B . lf X is an AW* -modelover A(B) (see[52]), then X is a B-cyclic Banach space.All the aforesaid leads to the following realization theorem. 5.4. Theorem [59]. The restricted descentof a complex Hilbert spacein the model V@) is an AW* -module over the algebra A(B) . Conversely,for any AW* -module X over A(B) there existsa unique (up to unitary equivalence) Hitbert space inside V@) whose restricted descent is unitarily equivalent to X. NONSTANDARD METHODS IN GEOMETRIC FUNCTIONAL ANALYSIS 99 5.5. Let X and Y be Banach-Kantorovich spaceswith norming lattices E and f' , respectively. A linear operator T: X -+ Y is said to be majorizable if there existsa positive operator ,S:E --+F suchthat lTxl < S(lxl) for all x e X .If E - F and S is an orthomorphism, a majorizableoperator is also called bounded, since in that situation 7" coincideswith the lowering from V@) of a bounded linear operator acting in Banach spaces.By interpreting Riesz-Schaudertheory in a Boolean-valuedmodel one arrives at a new concept of cyclic compactnessand obtains correspondingresults on the solvability of operator equations in Banach-Kantorovich spacest241. General majorizable operators have a far more complicated structure and their analysisrequires appeal to a considerablevariety of methods (see1241,1261, t 31l ) . 5.6. Banach-Kantorovich spacesand majorizable operators were first introduced by L.V. Kantorovich in [6]. It was he who proposedthe first applications to the solution of operator equations by the method of successive approximations (see U7l, tl9l). These objects possessa rich structure and have severalimportant applications in the area of spacesof measurable vector-valued functions and linear operators in such spacest261. In particular, the study of Banach-Kantorovich spacesleads to the notion of Banach spaceswith mixed norms, which is enormously useful in connection with the isometric classificationof Banach function spaces(see[26]). $6. Banach algebras 6.1. Certain classesof Banach algebrasyield some beautiful variations on the theme outlined in the previous section. Call a C*-algebra A a B- C* algebra if A is cyclic with respect to a Boolean algebra of projections B , where any projection in ^B is multiplicative, involutive and of unit norm. If A ts an AW* -algebraand .B a regular subalgebraof the Boolean algebraof central projections p(,4) , then A is a B- C.-algebra. We shatl therefore say that A is a ,B-AW' -algebraif .B is a regular subalgebraof ryc@). Now let A be a J B-algebraand ^B and p.(A) the same as before. lf A is a cyclic Banach space with respect to .B, we shall say that A is a B- J B-algebra. An isomorphism that commutes with the projections in B will be called a B-isomorphism. The following theorem, though in a slightly different form, was proved in 1671. 6.2. Theorem 1671.The restricted descentof a C* -algebra in the model V@) is a B- C* -algebra. Conversely,for every B- C. -algebra A , there exists inside V@) a unique (up to *-isomorphism) C* -algebra .il such that the restricteddescentof ,M H x- B-isomorphic to A. 6.3. Theorem. The restricted descentof an AW* -algebra (J B-algebra) from the model V@) is a B- AW* -algebra (B- J B-atgebra). Conversely,for any B- AW* -algebra (B- J B-algebra) A there existsa unique (to within an IOO A. G. KUSRAEV AND S. S. KUTATELADZE isomorphism) AW* -algebra (J B-algebra) ,M whoserestricteddescentis Bisomorphic to A . In addition, "{ witl be a factor in V@) if and only if B : 9r(A) . The formulated statementconcerning AW* -algebrasis obtained in l59l and [601. 6.4. The Boolean-valuedrealization of von Neumann algebras[66] is also worthy of mention. The above realization theorems form the foundation for Boolean-valuedanalysisof all theseclassesof Banachalgebras(see[59]-[62], [66], t67l). In particular, it was shown in [59] that for all infinite cardinals a and P there exists an AW* -algebrathat is simultaneously a- and Phomogeneous(a conjecture of Kaplansky in [52]). This fact is related to the location of cardinal numbersunder embeddingsin V@) (see[44], t68l). $7. Convexanalysis 7.1. The subdifferential is one of the most important conceptsin convex analysis (see [24], [28]). In this section, referring to a few examples,we shall show how to use Boolean-valuedanalysisto study the internal structure of subdifferentials.Take avector space X, d K-space E, and a sublinear operator P: X + E. The subdifferential AP of P at zero is also called the supporting set of P t281. By Gordon's theorem, we may assumethat E c 9 L so that we can "convert" P inside a suitable model into an ,9 valued sublinear operator, i.e., a sublinear functional. To be precise: 7.2. Theorem [54]. There exist a Banach space x and a continuoussublinear functional p: x -+ ,9 in the model V@) such that there is an isomorphic embedding of X into the Banach-Kantorovichspace x I with [(rX) f l,s d e n s e i nx \ - 1 and P:Fot.Inthissituation,forany U e?P thereis a unique element u e V@) for which [u e lpn: I and (J : tt J o l. The map U -- u is an affine isomorphismof the convex sets0 P and (0p) I . 7.3. Thus, the study of 0P largely reducesto that of 0p. For example, let us look at the extremal structure of the subdifferential 0P. Let Ch(P) denotethe set of extremepoints of 0 P. It shouldbe noted that by Theorem 2 the relations U e Ch(P) and [u e Ch(p)] - 1 are equivalent, and one can then use the classicalKrein-Mil'man Theorem and Mil'man's inversion of it for 0p. For a rigorous formulation of the result, we need some more definitions. The weak closure o- cl(Q) (cyclic hull mix(A)) is the set of all operators T e L(X , E) of the form Tx : o-hmTox (x e X) , where (7") is a net in C) (resp., Tx - o-lnrTrx (x e X) , where (f6) c O p(E)) partition and (26) is a of unity in . The weak cyclic closureof Cl is the set o-cl(mix(O)) . If o-clQ - Q or mix(Q) : Q, one saysthat O is weakly o-closedor cyclic, respectively. The definition of weak r-closedness is analogous. NONSTANDARD METHODS IN GEOMETRIC FUNCTIONAL ANALYSIS IOI 7.4. Theorem 1271,[281. Q) For any sublinear operator P: X -+ E the subdffirential coincides with the weakly cyclic closure of the convex hull of its extremepoints Ch(P) . (2) If P:X -> E is a sublinear operator and T € L(Y,X), then Ch(PoT)cCh(P) oI. A set A c L@, ^E) is operatorconvex(weaklybounded)if oQ+/Q c Q f o r a n y o , f e g I - , o * f : 1 ( t h e s e t{ T r : T € O } i s o r d e r b o u n d e d f o r all xeX). 7.5. Theorem 1241,t271. For a weakly bounded set Q c L(X , E) , the following assertionsare equivalent: (1) Cl - 0P for somesublinearP: X -. E; (2) Q rs convex, cyclic, and weakly r-closed; (3) Q rs convex, cyclic, and weakly o-closed; (4) Q rs operator convex and weakly o-closed. 7.6. Let Q: Z + E be a positive operator, P a sublinear operator from a vector space X to a K-space Z . The term disintegration in K-spaces refers to those parts of the calculus of subdifferentialsbased on the formula d(O o P) : O o 0P. This formula is not always true, but it is known to be valid if O is an order-continuous functional (E - R) . The general caseis analyzed with the help of Theorem 4.2. Let O , e , t be the same as tn 4.2. There exists an R^-sublinear operator p: Xn --+Z inside V@) for which - t o P (cf. a.3). From this and 7.2 weconcludethat pI "j OoP :Oo,-t o (roP) - e I op I oj - (g op) I oi, 0(OoP) : {u I oj:fu e 0(q p) : e o0pn- l]. " These argumentsyield the following result. 7.7. Theorem 1221.Let O be a positive order-continuous operator. The formula o(Oo P):Q-o0P is validfor any sublinearoperatorP if and only if A is a Maharam operator. 7.8. Further developmentsof disintegration in K-spacesmay be found in l24l and t281. On nonstandardmethods in convex analysisseealso [33], 1341, [36], and [54]. $8. Monadology 8.1. A central concept of infinitesimal analysisis the monad. According to Euclid's definition, "a monad is that through which the many becomeone." In the formal theory, a monad p(q) is definedas an externallist of the standard elementsof astandardfilter V,i.e., x e p(7; *-' (V't,Fe V)x e F. A syntactic characterizationof external setsthat are monads was proposed not long ago by Benninghofen and Richter [45]. It is useful to emphasizethat everv monad is a union of ultramonads-monads of ultrafilters. For such a IO2 A. G. KUSRAEVANDS. S. KUTATELADZE monad U the assertions(3x e U)rp(x) and (Vx)p@), where (p - e@) is an external formula, are equivalent. Hence it is clear that ultramonads are the genuine "elementary" objects of infinitesimal analysis. 8.2. For applications to the theory of operators, it is of essentialimportance to construct a synthetic theory in the framework of which both the nonstandard methods offered by Boolean-valuedmodels and external set theories can be used. Only preliminary results have so far been achieved in this direction; they pertain to the study of topological-type notions related with mixing-cyclic filters, topologies and so on, which play major roles in K-spaces. We shall point out one of the possible approachesto cyclic monadology. 8.3. Fix a standard complete Boolean algebra B and an external set A consisting of elements of a separatedBoolean-valued universe V@) . An element x e V@) is a member of the cyclic hull mix(,4) if and only if, for some internal family (or)rr= of elementsof ,,4 and an internal partition of unity (b)cr= in B, we have x - mix( eeb€x€.A monad p(V) is said to be cyclic if p(V) - mix p(,V). A point is said to be essentialif it lies in the monad of somepro-ultrafilter-a maximal cyclic filter or, more rigorously, 8n ultrafilter in V@) . 8.4. Theorem. (l) A standardfilter is cyclic if and only if its monad is cyclic. (2) A filter is extensionalif and only if its monad is the cyclic monadic hull of the set of its essentialpoints. As corollaries we cite the following Boolean-valuedanalogsof some classical criteria of A. Robinson. 8.5. Theorem. (l) A standard set is the lowering of a compactspaceif and only if each of its essentialpoints is near-standard. 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