nonstandard methods in geometric functional analysis · nonstandard methods in geometric functional...

Amer. Math. Soc. Transl.(2) vol. r5r, 1992

Nonstandard Methodsin Geometric Functional Analysis

UDC 517.11-517.98

A. G. KUSRAEV AND S. S. KUTATELADZE

Nonstandard methods in the modern sense consist of the explicit or im-plicit appeal to two different models of set theory-"standard" and "non-standard"-1s investigate concrete mathematical objects and problems. Themain development of such methods dates to the last thirty years, and theyhave now crystallized in several directions (see 1291, l42l and the bibliogra-phy cited there). The main directions are now known as infinitesimal andBoolean analysis. In this paper we shall outline new applications of non-standard methods to problems arising in the area of our personal interests,grouped together under the general heading 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 frequentlyreferred to by the expressive but rather unfortunate phrase "nonstandardanalysis"; nowadays one most frequently speaks of classical or Robinsoniannonstandard analysis. Infinitesimal analysis is characterized by the use of cer-tain conceptions, long familiar in the practice of natural sciences but frownedupon in twentieth-century mathematics, involving the notions of actual in-finitely large and infinitely small quantities.

1.2. Modern expositions of nonstandard analysis rely on formulas ofE. Nelson's internal set theory IST [58] and its later developments, the exter-nal set theories of K. Hrbacek (EXT) [49] and T. Kawai (NST) t53]. From thestandpoint of the "working mathematician-Philistine," the essence of thesetheories is as follows.

l99L Mathematics Subject Classffication. Pimary 46520, 03H05.

@ 1992 American Mathematical Society0065-9290192 $1.00 + $.25 per page

9 l

92 A. G. KUSRAEVANDS. S. KUTATELADZE

Ordinary mathematical objects and properties are called internal (andconsidered, if a rigorous formalization is desired, within the framework ofZermelo-Fraenkel set theory ZFC). One introduces a new predicate St(x) , ex-pressing the property of an object x to be standard (qualitatively speaking-obtained through existence and uniqueness theorems, i.e., the set of naturalnumbers is standard, but the infinitely large natural numbers are nonstan-dard). Mathematical formulas and concepts in whose construction the newpredicate is used will be called external. "Cantorian" sets endowed with ex-ternal properties are referred to as external. In Nelson's theory, such setsare considered only as terms of a metalanguage, which is used only for con-venience. In EXT and NST one can treat them as objects of Zermelo the-ory, which requires elaboration of a formalism and introduction of a newprimary predicate Int(x) , stating that the object x is internal. The avail-able formalisms ensure that the extension of ZFC is conservative, i.e., whenproving mathematical statements whose formulations do not involve externalconcepts, we may legitimately invoke the theories IST, EXT, and NST, as noless reliable than ZFC.

1.3. A point of crucial importance is that the new theories contain addi-tional rules, easily motivated at the intuitive level, which are known as theprinciples of nonstandard analysis. We present their rigorous formulationsin IST.

( 1) Transfer principle:

( v " " , ) " ' ( v " x r ) ( ( v " t r ) q ( x , x r x n ) - $ x ) p ( x , x t , . . . , x r ) ) ,

where p is an internal formula and rp - q(x , xr, ... , xn) (i.e., p does notcontain any free variables other than those listed).

(2) Idealization principle:

( v x r ) . . . l v x r ) ( v " o n r ) ( : x ) ( v y e 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) Standardization principle:

(vx r ) " ' ( vx , ) (v ' t x ) (3 ' t y ) (Y" t z )z e ! * z e x A q (2 , x t , . . . , xn ) ,

where e - e(z , xr, ... , xn) is an arbitrary formula. The index st indicatesthat the quantifier in question is relativized to standard sets; the index st finhas the analogous meaning with regard to standard finite sets.

$2. Boolean-valued analysis

2.1. Boolean-valued analysis is characterized by the extensive use of theterms lowering and lifting, cyclic hulls and mixing. The development ofthis trend, which emerged under the impetus of P. J. Cohen's remarkablework on the continuum hypothesis, leads to essentially new ideas and results,first and foremost, in the theory of Kantorovich spaces and von Neumann

NONSTANDARD METHODS IN GEOMETRIC FUNCTIONAL ANALYSIS 93

algebras. The modeling device offered by Boolean-valued analysis makesit possible, in particular, to consider the elements of functional classes asnumbers, which substantially facilitates the analysis and creates a uniquepossibility of automatically extending the scope of classical theorems.

2.2. The construction of a Boolean-valued model begins with a completeBoolean 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-valued universeV@) or class of ^B-sets:

v@) ._ u v:r,.aeO,

2.3. Taking an arbitrary formula of ZFC and interpreting the connectivesand quantifiers in the natural way in the Boolean algebra B, one defines itstruth value [,p\, which depends on the way in which A is built up fromatomicformulas x:! and x e y. Thetruthvaluesofthelatteraredefinedfor x, y e V@) by a recursion schema:

[ x e yn : - V yQ) n f i z : x \ ,z€dom(y)

l l x -yn : - V xQ)+ [ .2€yn^ A yQ)+ f i zex \z€dom(x) z€dom(y)

(the sign + symbolizes implication 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] - 1is valid, i.e., A is true inside V@) .

2.5. In the class V@) there is a natural equivalence x - !:- [x - ln :I , which preserves truth values. In this connection, one can use a specialdevice to go over to a separated universe 7@) , in which x : ! ++ l[x -

yn : 1 . In fact, the identification V@) :- 7@) is usually assumed withoutspecial mention. The basic properties of tr'(B) are expressed by the followingassertions.

2.6. Mixing principle. Let (b)er= be a partition of unity in .B , i.e., I #

4 -be Abr :0 , Vc r=bq- 1 . Forany fami ty ( " r )a .= o f theun ive rse V@)there exists a (unique) mixture of (xa)a6, with probabilities (b6)aE=, i.e., anelement x of the separated universe, denoted bt Dr.= brx, or sixrrrbrxl ,such that [x - xrn 2 b, for ( e E .

2.7 . Maximum principle.xo e V@) for which

For every formula e of ZFC there is an element

W=x)q(x)n : ne(xs)n.


In particular, V@) contains an object ,9 whichplays the role of the fieldR inside V@) .

2.8. Besides the above principles, there is an important procedure for pass-ing to V@) from the ordinary von Neumann universe V , where the latteris defined by the recursion schema

Vo i: {x: (38 e a)x € P(Vp)} , V :- U Vo.a € O n

This procedure is defined by the rule

on : - a , dom(xn) r - { /n ry e x } , im(xn) : - { l } .

The element x^ € V@) is known as the standard name of x . We thus havea canonical embedding of V into V@) . Apart from this we have a techniqueof lowerings and liftings of sets and correspondences.

2.9. Given an element x e V@), its lowering x I is defined by the rulex | :- {t e V@l: [/ e xn : l]. The set x J is cyclic, i.e., closed with respectto mixing of its elements.

2.10. Let F be a correspondence from X to I inside V@) . There existsa correspondence f' | -and it is unique- from X I to I I such that forany subset A 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-loweringf I : X - + Y I s u c h t h a t f J ( x ) - f ( x " ) ( x e X ) .

2 . l l . L e t x € P ( V @ ) ) . D e f i n e A l : - A a n d d o m ( x t ) - x , i m ( x J ) -

{1}. The element x f is called the lifting of x. It is easy to see that x 1lis the least cyclic set containing x,i.e., its cyclic hull: x lJ : mix(x) .

2 . l2 .Let X,Y e P1V@l) andlet F beacorrespondenceform X to Y.There exists a correspondence ̂ F | -and it is unique- from X I to Y Iinside V@) such that dom(,F 1) : dom(f') f and for every subset A ofdom(f') we have F I @ I) : F (A) 1 if and only if F is extensional, i.e.,

!1e F(x,) r l lxr : x . rn 3 V W1: !2n.YeF@r)

Inpart icular, amap f:X ---+Y J generatesafunction f I :X" --+ Y suchthat f t (x" ) - f (x) for x e X. I f necessary inspeci f iccases, thelower ingand lifting procedures can be iterated.

$3. Vector lattices

3.1. There are several excellent monographs on the theory of vector lattices

[4], [18], [19], [55], [70]. Vector lattices are also commonly known as Rieszspaces, and order-complete vector lattices as Kantorovich spaces or K-spaces.A K-space is said to be extended if any set of pairwise disjoint positive


elements in it has a supremum. The most important examples of extendedK-spaces are the following:

( 1) the space M (O , 2 , lt) of equivalence classes of measurable functions,where (O, I, 1r) is a measure space with lt a o-finite measure (or, moregenerally, a space with the direct sum property, see [18]);

(2) the space C*(Q) of continuous functions defined on an extremallydisconnected compact space O with values in the extended real line, takingthe values *oo only on a nowhere dense set [4], [9], [55];

(3) the space V of selfadjoint (not necessarily bounded) operators associ-ated with a von Neumann algebra (see [66]).

To save space, we shall restrict attention to the real case, since the analysisof complex K-spaces is entirely analogous. The symbol P(E) will denote theBoolean algebra of order projections in a K-space E . If E contains an orderunit, C(E) is the Boolean algebra of unit elements (fragments of the identity)in E. The algebras f@) and €(,8) are isomorphic and known as the baseof E. Throughout the sequel, B will be a fixed complete Boolean algebra.The basis for Boolean-valued analyris of vector lattices is the following result.

3.2. Theorem (Gordon [6]). Let ,9 be the field of real numbers in themodel V@) . The algebraic system .q I Q.e., .q with lowered operationsand order) is an extended K-space. Moreover, there exists an isomorphism Xof the Boolean algebra B onto the base ry@) such that

b < [ x : Y \ , - x (b )x - x (b ) ! ,b < f ix < yn *- x(b)x < x(b)y

f o r a l l x , y e . q I a n d b e B .

Throughout the sequel, R will denote the field of real numbers insideV@) . If the base of a K-space E is isomorphic to .8, then E itself isisomorphic to the foundation Eo C.q l, and in this situation ̂ E is extendedif and only if E0 - SE J . Under these circumstances one says that ,% J is amaximal extension and I a Boolean-valued realization of the K-space .E .It is noteworthy that Boolean-valued realizations of certain structures lead tosubsystems of the field I .

3.3. Theorem [25].(l) A subgroup of the additive group of ,q is a Boolean-valued realization

of an archimedean lattice-ordered group.(2) A vector sublattice of ,q , considered as a vector lattice over the field R^

is a Boolean-valued realization of an archimedean vector lattice.(3) An archimedean f-ring contains two mutually complementary compo-

nents, one of which is a group with zero multiplication realized as in (l), andthe other has a subring of the ring ,9 as a Boolean-valued realization.

(Q An archimedean f -algebra contains two mutually complementary com-ponents, one of which is a vector lattice with zero multiplication realized as in


(2), and the other is realized as a subring and sublattice of ,q , considered asan 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-spaces and functional calculus. Let Qbe a Stonean compact subspace of the Boolean algebra B and define C*(Q)as in 3.1 (2). We call a map e:R -- B a resolution of 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 al lresolut ionsof unityin ^8. Thesets C."(Q)and ^B(R) can be endowed canonically with the structure of an extended K-space (see [4] and [19]).

3.5. Theorem (1291, t50l). The extended K-space g I is (algebraicallyand order) isomorphic to each of the K-spaces B(R) and C*(q . Underthis isomorphism an element x €. g I is mapped onto a resolution of unityt - el (l e R) and onto afunction 7:Q +R by the formulas

el :- [x < l"n (r e R),7(q) : - in f { / e R: [x < /nn e q] (a e 91.

3.6. Let gR and ,% (R) be the o-algebra of Borel sets and the vectorlattice of Borel functions, respectively, on the real line. We identify B withthe 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-continuousBoolean homomorphism F:9n - B) such that tt(-x , t) : el (l e n; .The measure p defines an integral

I *(f ) :- [ f Al drt(t) (f e ,q 8)).- - JR

In this situation lr(f ) is the unique element of !t I for which

[ . l ' , ( f ) < rnn :p ( f< / ) ) .

3.7. Theorem (1291, [50]). The map I*:,9(R) + .9/ ! is the unique se-quentially o-continuous lattice and algebraic homomorphism satisfying thecondition

/"( id^) - x.

3.8. For other aspects of Boolean-valued analysis of vector lattices, see [7],[8], [24], 1291, [50], [51], t651.

$4. Positive operators

4.1. General information about positive and order-bounded operators maybe found in 1241, 1291. Take arbitrary K-spaces Z and E. A positive oper-ator O: Z - ^E will be called a Maharam operator if it is order continuousand O([0 , z l ) : [0 , O(z) ] for every z e Z- , where la , b l ; - {c : a 1c < b}is an order interval. Let mZ be a maximal extension of Z and D(O)* the


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 extends toa Maharam operator on the whole of D(O) . We say that O is essentiallypositive if O > 0 and O(lrl) - 0 implies z - 0.

4.2. Theorem t221. Let Q be an essentially positive Maharam operator.There exist a K-space Z and an essentially positive o-continuous functionale: Z + 9 in the model V@) , and there exists an isomorphism t from D(O)onto the K-space Z I such that Q: e I o t.

4.3. The above result reduces the investigation of Maharam operators toanalysis of the class of o-continuous positive functionals. What is the situ-ation with regard to arbitrary positive operators? Various approaches basedon Boolean-valued analysis may be adopted here. Let us consider an order-bounded operator from a vector lattice Z into E:9 l. There exists anorder-bounded R^-linear functional e: Zn -f R inside V@) for whichQ: g l o / , where j :z -- z^ (z e Z). The map Q -- rp is anjso-morphism of the space of all order-bounded operators L,(2, ^E) onto Z Lwhere 7 it the space of order-bounded function als on Z . In particular,O > 0 if and only if np > 0n - I . The disadvantage of this device is thatthe map O + p does not preserve order-continuity.

On the other hand, for a positive operator Q: Z + E one can constructan essentially positive Maharam operator O and a lattice homomorphismh:Z - D(iD) such that Q: @o h,where the pair (h,6) is minimal in acertain sense (see [1]). Appealing to Theorem 4.2, we obtain a representationQ:e Ior ' ,where t ' : -hot and g isanessent ia l lyposi t ive o-cont inuousfunctional in the model V@) . The disadvantage of this approach is that thespace D(lD-) may prove to be invisible. However, in a fairly general situation,D(lD) is realized as the space of functions (in two variables) on P * Q, whereP and O are Stonean compact subspaces of Z and ,8, respectively (see

t5 5l).4.4. The above arguments are easily applied to the algebra of fragments of

an arbitrary positive operator O acting from a vector lattice Z to a K-spaceE with filter of units 6 and base f@) (see [] and t39l). We dwell on therepresentation of the projection S of an operator Z onto the component

{Afo generated by the operator O. Let us call a set of operutors g inL,(Z , E) a generating set if Ox- - sup{p@x:p € g} for all x e Z. Tostudy interesting fragments by lifting into a Boolean-valued universe, one canreduce everything to the case of functionals. For the latter, using infinitesimalrepresentations, one readily proves that

Sx- inf* {" pTx; pd Qx ! o , p e ,q} ,

S x - i n f { " T y , O ( x - y ) = 0 , 0 < y ( x } ,

where * is the standardization symbol, o

the "standard part" operation, r


denotes infinite smallness and -.. denotes the exactness of the formula, i.e.,the attainability of equality.

Interpreting the above nonstandard representations and performing thelowering, one arrives at the following formulas l29l:

Sx - supinf {zTr* no Tr ,0 < y I x , n eg(E) , n i l (x - y) < e} ,e€€

Sx : sup in f { ( TEp)d Tr :pDr 1 e , p e .q , f t q_ f (E ' ) } .e€(

$5. Banach-Kantorovich spaces

5.1. A Banach-Kantorovich space consists of a (real or complex) vectorspace X,d K-space E,andavectornonn l . l :X + E suchthatthefol lowingconditions hold: (1) the nonn is decomposable, i.e., if lxl - et+e2, wherex 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 rsuitable x, x2 e X; (2) X is o-complete, i.e., for any net (x") c X , ifo - l im lxo - *p l :0 , then o - l im lxo - x l : 0 fo r some x e X . We sha l l

assume that { lxl :x e X}od - E c.q I. l f E is extended, i .e., E :,9 Lthen X is also said to be extended. An example of an extended Banach-Kantorovich space is the space M(O,Z,lt,Y) of (equivalence classes of)strongly measurable vector-valued functions with values in a Banach spaceY .

5.2. Theorem 1231. Let x be a Banach space in the model V@) . Thenthe lowering x ! is an extended Banach-Kantorovich space. Conversely, if Xis an extended Banach-Kantorovich space, there exists a unique (up to linearisometry) Banach space x in V@) whose lowering is linearly isometric to X .

5.3. Let us call the bounded part of the space x I the restricted descent ofx. The restricted descents of Banach spaces in V@) constitute the class of.B-cyclic Banach spaces. Let B be the complete Boolean algebra of norm oneprojections in a Banach space X. We shall say that X is cyclic with respectto B, or .B-cyclic, if, for an arbitrary partition of unity (nr)rr= c B and anybounded family ("r)r.- c X there exists a unique element x e X such thatnexc: ftex (( e E) and llxll < trp(.= 11x6ll . Let A(B) denote an arbitrarycommutative AW* -algebra whose complete Boolean algebra of idempotentsis isomorphic to B . lf X is an AW* -model over A(B) (see [52]), then X isa B-cyclic Banach space. All the aforesaid leads to the following realizationtheorem.

5.4. Theorem [59]. The restricted descent of a complex Hilbert space in themodel V@) is an AW* -module over the algebra A(B) . Conversely, for anyAW* -module X over A(B) there exists a unique (up to unitary equivalence)Hitbert space inside V@) whose restricted descent is unitarily equivalent toX .


5.5. Let X and Y be Banach-Kantorovich spaces with norming latticesE and f' , respectively. A linear operator T: X -+ Y is said to be majoriz-able if there exists a positive operator ,S:E --+ F such that lTxl < S(lxl) forall x e X .If E - F and S is an orthomorphism, a majorizable operatoris also called bounded, since in that situation 7" coincides with the loweringfrom V@) of a bounded linear operator acting in Banach spaces. By inter-preting Riesz-Schauder theory in a Boolean-valued model one arrives at anew concept of cyclic compactness and obtains corresponding results on thesolvability of operator equations in Banach-Kantorovich spaces t241. Gen-eral majorizable operators have a far more complicated structure and theiranalysis requires appeal to a considerable variety of methods (see 1241,1261,t3 1 l ) .

5.6. Banach-Kantorovich spaces and majorizable operators were first in-troduced by L.V. Kantorovich in [6]. It was he who proposed the firstapplications to the solution of operator equations by the method of succes-sive approximations (see U7l, tl9l). These objects possess a rich structureand have several important applications in the area of spaces of measurablevector-valued functions and linear operators in such spaces t261. In partic-ular, the study of Banach-Kantorovich spaces leads to the notion of Banachspaces with mixed norms, which is enormously useful in connection with theisometric classification of Banach function spaces (see [26]).

$6. Banach algebras

6.1. Certain classes of Banach algebras yield some beautiful variations onthe 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. IfA ts an AW* -algebra and .B a regular subalgebra of the Boolean algebra ofcentral projections p(,4) , then A is a B- C.-algebra. We shatl therefore saythat A is a ,B- AW' -algebra if .B is a regular subalgebra of ry c@). Now letA be a J B-algebra and ^B and p.(A) the same as before. lf A is a cyclicBanach 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 aB-isomorphism. The following theorem, though in a slightly different form,was proved in 1671.

6.2. Theorem 1671. The restricted descent of a C* -algebra in the modelV@) is a B- C* -algebra. Conversely, for every B- C. -algebra A , there existsinside V@) a unique (up to *-isomorphism) C* -algebra .il such that therestricted descent of ,M H x- B-isomorphic to A.

6.3. Theorem. The restricted descent of an AW* -algebra (J B-algebra)

from the model V@) is a B- AW* -algebra (B- J B-atgebra). Conversely, forany B- AW* -algebra (B- J B-algebra) A there exists a unique (to within an

IOO A. G. KUSRAEV AND S. S. KUTATELADZE

isomorphism) AW* -algebra (J B-algebra) ,M whose restricted descent is B-isomorphic to A . In addition, "{ witl be a factor in V@) if and only ifB : 9r(A) . The formulated statement concerning AW* -algebras is obtainedin l59l and [601.

6.4. The Boolean-valued realization of von Neumann algebras [66] is alsoworthy of mention. The above realization theorems form the foundation forBoolean-valued analysis of all these classes of Banach algebras (see [59]-[62],[66], t67l). In particular, it was shown in [59] that for all infinite cardinalsa and P there exists an AW* -algebra that is simultaneously a- and P-homogeneous (a conjecture of Kaplansky in [52]). This fact is related to thelocation of cardinal numbers under embeddings in V@) (see [44], t68l).

$7. Convex analysis

7.1. The subdifferential is one of the most important concepts in convexanalysis (see [24], [28]). In this section, referring to a few examples, weshall show how to use Boolean-valued analysis to study the internal structureof subdifferentials. Take avector space X, d K-space E, and a sublinearoperator P: X + E. The subdifferential A P of P at zero is also calledthe supporting set of P t281. By Gordon's theorem, we may assume thatE 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 continuous sublin-ear functional p: x -+ ,9 in the model V@) such that there is an isomorphicembedding of X into the Banach-Kantorovich space x I with [(rX) f l,sd e n s e i n x \ - 1 a n d P : F o t . I n t h i s s i t u a t i o n , f o r a n y U e ? P t h e r e i sa unique element u e V@) for which [u e lpn: I and (J : tt J o l. Themap U -- u is an affine isomorphism of the convex sets 0 P and (0 p) I .

7.3. Thus, the study of 0P largely reduces to that of 0p. For example,let us look at the extremal structure of the subdifferential 0P. Let Ch(P)denote the set of extreme points of 0 P. It should be noted that by Theorem2 the relations U e Ch(P) and [u e Ch(p)] - 1 are equivalent, and onecan then use the classical Krein-Mil'man Theorem and Mil'man's inversionof it for 0p. For a rigorous formulation of the result, we need some moredefinitions. The weak closure o- cl(Q) (cyclic hull mix(A)) is the set ofall 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 Oand (26) is a partition of unity in p(E)) . The weak cyclic closure of Cl isthe set o-cl(mix(O)) . If o-clQ - Q or mix(Q) : Q, one says that O isweakly o-closed or cyclic, respectively. The definition of weak r-closednessis analogous.

NONSTANDARD METHODS IN GEOMETRIC FUNCTIONAL ANALYSIS IOI

7.4. Theorem 1271, [281. Q) For any sublinear operator P: X -+ E thesubdffirential coincides with the weakly cyclic closure of the convex hull of itsextreme points Ch(P) .

(2) If P:X -> E is a sublinear operator and T € L(Y,X), thenC h ( P o T ) c C h ( P ) o I .

A set A c L@, ^E) is operator convex (weakly bounded) if oQ+/Q c Qf 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 ra l l x e X ) .

7.5. Theorem 1241, t271. For a weakly bounded set Q c L(X , E) , the

following assertions are equivalent:(1) Cl - 0P for some sublinear P: 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 froma vector space X to a K-space Z . The term disintegration in K-spacesrefers to those parts of the calculus of subdifferentials based on the formulad(O o P) : O o 0P. This formula is not always true, but it is known to bevalid if O is an order-continuous functional (E - R) . The general case isanalyzed 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 whichp I "j

- t o P (cf. a.3). From this and 7.2 weconclude that

O o P : O o , - t o ( r o P ) - e I o p I o j - ( g o p ) I o i ,

0 ( O o P ) : { u I o j : f u e 0 ( q " p ) : e o 0 p n - l ] .

These arguments yield 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 sublinear operator P if and onlyif A is a Maharam operator.

7.8. Further developments of disintegration in K-spaces may be found in

l24l and t281. On nonstandard methods in convex analysis see also [33], 1341,[36], and [54].

$8. Monadology

8.1. A central concept of infinitesimal analysis is the monad. According toEuclid's definition, "a monad is that through which the many become one." Inthe formal theory, a monad p(q) is defined as an external list of the standardelementsof astandardfi l ter V,i .e., x e p(7; *- ' (V't ,F e V)x e F. Asyntactic characterization of external sets that are monads was proposed notlong ago by Benninghofen and Richter [45]. It is useful to emphasize thateverv 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@) isan external formula, are equivalent. Hence it is clear that ultramonads arethe genuine "elementary" objects of infinitesimal analysis.

8.2. For applications to the theory of operators, it is of essential impor-tance to construct a synthetic theory in the framework of which both thenonstandard methods offered by Boolean-valued models and external set the-ories can be used. Only preliminary results have so far been achieved inthis direction; they pertain to the study of topological-type notions relatedwith mixing-cyclic filters, topologies and so on, which play major roles inK-spaces. We shall point out one of the possible approaches to cyclic mon-adology.

8.3. Fix a standard complete Boolean algebra B and an external set Aconsisting of elements of a separated Boolean-valued universe V@) . Anelement x e V@) is a member of the cyclic hull mix(,4) if and only if, forsome internal family (or)rr= of elements of ,,4 and an internal partition ofunity (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 essential if it lies in the monad of some pro-ultrafilter-amaximal 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 extensional if and only if its monad is the cyclic monadic hull

of the set of its essential points.

As corollaries we cite the following Boolean-valued analogs of some clas-sical criteria of A. Robinson.

8.5. Theorem. (l) A standard set is the lowering of a compact space if andonly if each of its essential points is near-standard.

(2) A standard set is the lowering of a totally bounded space if and only ifeach of its essential points is pre-near-standard.

RnrnnrNcns

l. G. P. Akilov, E. V. Kolesnikov, and A. G. Kusraev, On the order-continuous extension ofa positive operator, Sibirsk. Mat. Zh. 29 ( 1988), no. 5, 24-35; English transl. in Siberian Math.J .29 (1988 ) , no . 5 .

2. K. I. Beidar and A. V. Mikhalev, Orthogonal completeness and algebraic systems, UspekhiMat. Nauk 40 (1985), no. 6, 79-l15; English transl. in Russian Math. Surveys 40 (1985), no. 6.

3. P. Vopenka, Mathematics in the Alternative Set Theory, Teubner, Leipzig, 1979.4. B. Z. Vulikh, Introduction to the theory of partially ordered spaces, Fizmatgiz, Moscow,

l96l; English transl. Noordhoof, Groningen, 1967.5. R. Goldblatt, Topoi: The Categorial Analysis of Logic, North-Holland, Amsterdam, 1979.6. E. I. Gordon, Real numbers in Boolean-valued models of set theory and K-spaces, Dokl.

Akad. Nauk SSSR 237 (1977), no. 4, 773-775 English transl. in Soviet Math. Dokl. l8 (1977).7. -, K-spaces in Boolean-valued models of the theory of sets, Dokl. Akad. Nauk SSSR

258 (1981), no. 4, 777-780; English transl. in Soviet Math. Dokl. 23 (1981).8. -, On theorems on the presemation of relations in K-spaces, Sibirsk. Mat. Zh. 23

(1982), no. 3, 55-65; English transl. in Siberian Math. J.23 (1982), no. 3 (1983).

NONSTANDARD METHODS IN GEOMETRIC FUNCTIONAL ANALYSIS IO3

9. -, Nonstandardfinite-dimensional analogs of operators in t,r(R'), Sibirsk. Mat. Zh.29 (1988), no.2, 45-59; English transl. in Siberian Math. J.29 (1988), no. 2.

10. -, Relatively standard elements in E. Nelson's internal set theory, Sibirsk Mat. Zh. 30(1989), no. l, 89-95; English transl. in Siberian Math. J.30 (1989), no. 1.

ll. E. I. Gordon and V. A. Lyubetskii, Some applications of nonstandard analysis in thetheory of Boolean-valued measures, Dokl. Akad. Nauk SSSR 256 (l9Sl), no. 5, 1037-1041;English transl. in Soviet Math. Dokl. 23 (1981).

12. A. K. Zvonkin and M. A. Shubin, Nonstandard analysis and singular perturbations ofordinary diferential equations, Uspekhi Mat. Nauk 39 (1984), no.2,77-127; English transl. inRussian Math. Surveys 39 (1984), no. 2.

13. M. Davis, Applied Nonstandard Analysis, Wiley-Interscience, New York, 1977.14.T. Jech, Lectures in Set Theory with Particular Emphasis on Forcing, Springer, Berlin,

197 t .15. V. G. Kanovei, Correctness of the Euler method of decomposing the sine function into an

infinite product, Uspekhi Mat. Nauk 43 (1988), no.4, 57-81; English transl. in Russian Math.Surveys 43 (1988), no. 4.

16. L. V. Kantorovich, Toward a general theory of operations in semiordered spaces, Dokl.Akad. Nauk SSSR f ( 1936), no. 7 , 27 l-27 4. (Russian)

17. -, On a class of functional equations, Dokl. Akad. Nauk SSSR 4 (1936), no. 45, 2Il-216. (Russian)

18. L. V. Kantorovich and G. P. Akilov, Functional Analysis, "Nauka", Moscow, 1977; En-glish transl., Pergamon Press, Oxford-New York, 1982.

19. L. V. Kantorovich, B. Z. Vulikh, and A. G. Pinsker, Functional analysis in partiallyordered spaces, GITTL, Moscow, 1950. (Russian)

20. P. J. Cohen, Set Theory and the Continuum Hypothesis, Benjamin, New York, 1966.21. A. G. Kusraev, Some categories and functors of Boolean-valued analysis, Dokl. Akad.

Nauk SSSR 271 (1983), no. l, 281-284; English transl. in Soviet Math. Dokl. 28 (1983).22. -, Order-continuous functionals in Boolean-valued models of set theory, Sibirsk Mat.

2h.25 (1984), no. 1, 69-79; English transl. in Siberian Math. J.25 (1984), no. 1.23. -, Banach-Kantorovich spaces, Sibirsk Mat.2h.26 (1985), no.2, 119-126; English

transl. in Siberian Math. J.26 (1985), no. 2.24. -, Vector Duality and its Applications, "Nauka", Novosibirsk, 1985. (Russian)25. -, Boolean-valued models and ordered algebraic systems, in: Eighth All-Union Con-

ference on Mathematical Logrc. Abstracts of Lectures, Moscow, 1986, 99. (Russian)26. -, Linear operators in lattice ordered spaces, in: Studies in Geometry in the Large and

Mathematical Analysis, "Nauka", Novosibirsk, I 987, 84-123. (Russian)27. A. G. Kusraev and S. S. Kutateladze, Analysis of subdffirentials by means of Boolean

valued models, Dokl. Akad. Nauk, SSSR 265 (1982), no. 5, 1061-1064; English transl. in SovietMath. Dokl. 26 (198211983).

28. -, Subdffirential Calculu.s, "Nauka", Novosibirsk, 1987. (Russian)29. A. G. Kusraev and S. A. Malyugin, Atomic decomposition of vector measureg Sibirsk Mat.

2h.30 (1989), no. 5, 101-110; English transl. in Siberian Math. J.30 (1989), no. 5.31. A. G. Kusraev and V. Z. Strizhevskll, Lattice normed spaces and majorized operators,

Studies in Geometry and Analysis, "Nauka", Novosibirsk, 1986, 56-102. (Russian)32. S. S. Kutateladze, Descents and Ascents, Dokl. Akad. Nauk SSSR 272 (1983), no. 3,

521-524; English transl. in Soviet Math. Dokl. 28 (1983).33. -, Infinitesimal tangent cones, Sibirsk. Mat.2h.26 (1985), no. 6, 67-76; English

transl. in Siberian Math. J.26 (1985), no. 6.34. -, Nonstandard methods in subdffirential calculus, Partial Differential Equations,

"Nauka", Novosibirsk, 1986, pp. 116-120. (Russian)35. -, Cyclic monads and their applications, Sibirsk Mat. Zh. 27 (1986), no. l, 100-l l0;

English transl. in Siberian Math. J.27 (1986), no. l.36. -, A variant of nonstandard convex programming Sibirsk Mat.2h.27 (1986), no. 4,

84-92; English transl. in Siberian Math. J.27 (1986), no. 4.37. -, Directions of nonstandard analysis, Trudy Inst. Mat. (Novosibirsk) 14 (1989), 153-

182. (Russian)

t04 A. G. KUSRAEV AND S. S. KUTATELADZE

38. -, Monads of pro-ultrafilters and of extensional filters, Sibirsk Mat.2h.30 (1989), no.I,129-132; English transl. in Siberian Math. J. 30 (1989), no. l.

39. -, Components of positive operators, Sibirsk Mat.2h.30 (1989), no. 5, lll-119;English transl. in Siberian Math. J. 30 (1989), no. 5.

40. V. A. Lyubetskii and E. I. Gordon, Boolean extensions of uniform structures, Studies inNonclassical Logics and Formal Systems, "Nauka", Moscow, 1983, pp. 82-153. (Russian)

41. V. A. Lyubetskil, On some algebraic questions of nonstandard analysis, Dokl. Akad. NaukSSSR 280 (1985), no. 1,38-41; English transl. in Soviet Math. Dokl.3f (1985).

42. S. Albeverio, R. Hoegh-Krohn, J. E. Fenstad, and T. Lindstrom, Nonstandard Methodsin Stochastic Analysis and Mathematical Physics, Academic Press, Orlando, FL, 1986.

43. W. A. J. Luxemburg (ed.), Applications of Model Theory to Algebra, Analysis and Proba-bility, Holt, Rinehart and Winston, New York, 1969.

44. J. L. Bell, Boolean Valued Models and Independence Proofs in Set Theory, ClarendonPress, Oxford, 1979.

45. B. Benninghofen and M. M. Richter, A general theory of superinfinitesimals, Fund. Math.128 (1987), no. 3, 199-215.

46. C. Cristiant, Der Beitrag Gridels frir die Rechfertigang der Leibnizschen ldee von der In-finitesimalen, Osterreich. Akad. Wiss. Math. Nachr. f92 (1983), nos. 1-3,25-43.

47 . W. Henson and J. Keisler, On the strength of nonstandard analysis, J. Symbolic Logic 51(1986) , no.2 ,377-386.

48. R. B. Holmes, Geometric Functional Analysis and its Applications, Springer, Berlin-NewYork, 1975.

49. K. Hrbacek, Nonstandard set theory, Amer. Math. Monthly 86 (1979), no. 8, 659-677.50. T. Jech, Abstract theory of abelian operator algebras: An application of forcing, Trans.

Amer. Math. Soc. 289 (1985), no. 1, 133-162.51. -, First order theory of complete Stonean algebras, Canad. Math. Bull. 30 (1987), no.

4,385-392.52. I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), no. 4, 839-858.53. T. Kawai, Axiom systems of nonstandard set theory, Logic Symposia, Hakone, 1979, 1980,

Springer, Berlin-New York, 1981, pp. 57-65.54. A. G. Kusraev, Boolean valued convex analysis, Mathematische Optimierung. Theorie

und Anwendungen, Wartburg/Eisenach, 1983, pp. 106-109.55. W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces, vol. l, North-Holland, Amsterdam-

London, 1971.56. D. Maharam, On kernel representation of linear operators, Trans. Amer. Math. Soc. 70

(1955), no. l , 229-255.57. E. Nelson, Radically Elementary Probability Theory, Princeton University Press, Prince-

ton, N.J., 1987.58. -, The syntax of nonstandard analysis, Ann. Pure Appl. Logic 38 (1988), no.2,123-

r34.59. M. Ozawa, A classffication of type I AW* -algebras and Boolean valued analysis, J. Math.

Soc. Japan 36 (1984), no. 4, 589-608.60. -, A transfer principle from von Neumann algebras to AW* -algebras, J . London Math.

Soc. (2) 32 (1985) , no. l , l4 l -148.61. -, Boolean valued approach to the trace problem of AW" -algebra.s, J. London Math.

Soc. (2) 33 (1986) , no.2 ,347-354.62. -, Embeddable AW'-algebras and regular completions, J. London Math. Soc. (2) 34

(1986 ) , no . 3 , 5 l l - 523 .63. A. Robinson, Non-standard Analysis, North-Holland, Amsterdam-London, 1970.64. R. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin's problem, Ann.

of Math. 94 (1972), no.2,201-245.65. G. Takeuti, Two Applications of Logic to Mathematics,Ivanami & Princeton University

Press, Tokyo-Princeton, N.J., I 978.66. -, Von Neumann algebras and Boolean vahted analysis, J. Math. Soc. Japan 35 ( 1983),

no . l . l - 21 .67. -, C* -algebras and Boolean valued analysis, Japan J. Math. 9 (1983), no.2,207-246.

NONSTANDARD METHODS IN GEOMETRIC FUNCTIONAL ANALYSIS I05

68. G. Takeuti and W. M. Zaing, Axiomatic Set Theory, Springer, New York, 1973.69. D. Tall, The calculus of Leibniz-an alternative modern approach, Math. Intelligencer 2

(1979180), no. 1, 54.70. A. Z. Zaanen, Riesz Spaces, Vol. II, North-Holland, Amsterdam, 1983.

Translated bv D. LOUVISH

nonstandard methods in geometric functional analysis · nonstandard methods in geometric functional...

Documents