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Владикавказский математический журнал Январь-март, 2000, Том 2, Выпуск 1
YJ\K 517.11+517.98
ON COMBINED NONSTANDARD METHODS IN FUNCTIONAL ANALYSIS
A. G. Kusraev and S. S. Kutateladze
The main nonstandard tool-kits are known as infinitesimal analysis (Robinson's nonstandard analysis) and Boolean-valued analysis. Sharp distinctions between these two versions of nonstandard analysis in content and technique notwithstanding, many ways are open to their simultaneous application. One of the simplest approaches consists in successive application of different nonstandard methods. It is demonstrated that combining is often useful in settling the problems of functional analysis which stem mainly from the theory of vector lattices.
0. Introduction
The main nonstandard tool-kits are known as infinitesimal analysis (Robinson's nonstandard analysis) and Boolean-valued, analysis (see [1] and the references therein). Sharp distinctions between these two versions of nonstandard analysis in content and technique notwithstanding, many ways are open to their simultaneous application [2, 3]. One of the simplest approaches consists in successive application of different nonstandard methods [4, 5]. A few examples below demonstrate that combining is often useful in settling the problems of functional analysis which stem mainly from the theory of vector lattices.
We believe firmly in importance of constructing a synthetic theory that utilizes both nonstandard ways of mathematical reasoning.
1. Boolean-valued models
The Boolean-valued approach is less popular than its infinitesimal counterpart. It so stands to reason to sketch the former. The reader may find the relevant details in [1, 6]. A Boolean-valued model V® is constructed from a complete Boolean algebra ®. The class V® consists of B-valued functions and its elements are thought as B-valued sets. To a set-theoretic proposition S := S(xi,..., xn) with parameters x\,...,xn E there is assigned its Boolean truth value [5] E ®. The proposition S holds true in if [5] = 1. The following result asserts that is a Boolean-valued model of ZFC, the Zermelo-Fraenkel Set Theory.
© 2000 Kusraev A. G. and Kutateladze S. S.
1.1. Boolean-valued transfer principle. For every theorem S(xi,----xn)
of ZFC, the equality ..., xn)J = 1 holds, i.e., S(x i,...,xn) is valid inside
1.2. Maximum principle. For every formula S(v, xi,..., xn) (xi,... ,xn E y(®)) of ZFC with one free variable v there exists an element x E F® for which
[(3u)5(u, x1,..., a;n)] = [S(>, xx,..., a:n)].
In particular, contains some object 71 that plays the role of the reals R inside y (®).
1.3. Restricted transfer principle. There is an important procedure for passing from the conventional von Neumann universe
V to V(®). Proceed as follows: Given ./■ E V. denote by the symbol x the standard name of x in Vi.e., the element determined from the following recursion schema:
0:= 0, dom (x) := {y \ y E x}, im(i) := 1.
For every restricted formula S(v i,..., vn) and elements x-\_,...,xn E F® the equivalence holds:
S(x!,..., xn) [5(ii,... ,xn)J = 1.
1.4. Descent functor. There is a functorial procedure, called descent and denoted by (•),),, which assigns to each mathematical object inside a Boolean-valued model a standard object of a similar type. Given an element x E
, its descent
is defined by x^:= {t E V® [i 6 x] = 1}. The descent .r! of .r is a set. If .V.)'. / e V^ and / is a mapping from X to Y inside V® then fl is a mapping from A'.! to ) The same is true for a relation inside V®. Therefore, the descent of an algebraic structure is again an algebraic structure of a similar type.
2. Boolean-valued integrals
Necessary information from the theory of vector lattices can be found in [10]. The following result is basic for what follows, see [1, 7].
2.1. Gordon's Theorem. Let 7Z be the reals inside The algebraic
system 711 iS a universally complete vector lattice. Moreover, there exists an isomorphism x of ® onto the Boolean algebra of band projections in 711 su°h that
b<{x = y}^ x{b)x = x{b)y, b<fx <y}^ x{b)x < x{b)y
for all x, y E 7^4- and b EM.
Moreover, every order complete vector lattice F is isomorphic to an order dense ideal of the universally complete vector lattice 1ZJ,. Denote such isomorphism by iF.
We now consider the class of the so-called Maharam operators. Let E and F be vector lattices. A linear operator <l> : /•,' —/•' is a Maharam opera,tor if $ is positive (0 < .r e E => 0 < <!>./•). order-interval preserving ( <!»( J), .r] = J). <!>./•]. 0 < .r e F). and order continuous (xa \ 0 => ®xa \ 0). The following result gives a Boolean-valued interpretation for a Maharam operator [10].
2.2. Theorem. Given order complete vector lattices E and /•'. lei <l> : /•,' —/•' be a strictly positive Maharam operator. In a suitable Boolean-valued model there are an order complete vector lattice £ and a strictly positive order continuous functional (p : £ —1Z such that the diagram commutes
E
$
F
IE
IE
+ ni
i.e., if ° $ = ip J, ouE for a suitable order continuous lattice monomorphism ie :
This result reduces the study of the class of Maharam operators to that of the class of order continuous positive functionals. At the same time every order continuous linear functional is representable as Lebesgue integral over some measure space. We may thus conclude that the essential part of the theory of Maharam operators is simply a model-theoretic modification of the classical integration theory. The former emerges from the latter on using the Boolean-valued machinery.
The Loeb measure is proved to be a very useful tool of nonstandard analysis. This construction extends without difficulties to measures with values in a Ba-nach space. Extension becomes more involved for measures with values in a vector lattices without a distinguished norm. Further progress in this direction requires developing infinitesimal constructions in a Boolean-valued model.
2.3. Problem. Suggest a Boolean-valued version of the Loeb measure and integration technique. Study the corresponding class of linear operators.
3. Atomic decomposition of vector measures
We now demonstrate that successive application of the two nonstandard approaches is effective sometimes. We consider the problem of atomic decomposition of a finitely additive measure. Let A be a Boolean algebra and let F be a vector lattice. By a vector measure we mean an arbitrary mapping ¡j, : A —F such that
[j,(ai V a2) = [J,(a 1) + p(a2) for all disjoint a,i,a,2 E >4. Denote by ba (*4, F) the space of all bounded F-valued measures and put ba (A) := ba (*4,1Z). A measure p E ba (*4, F) is positive if p{a) > 0 for all a E A. It is well known that ba (A, F) is a Kantorovich space whose positive cone coincides with the set of positive measures. An atom of a measure p E ba (A) is an element a0 E A such that p(a0) # 0 and for every a E A, a < ao, either {¿(a) = 0, or p(a0 \ a) = 0. We say that a measure p is atomic if p belongs to the band in ba (A) generated by discrete elements, and p is diffuse if p belongs to the complementary band.
The infinitesimal approach uses Loeb's [11] concept of infinitely fine partition. By a finite partition in a Boolean algebra A, we mean a finite collection {ai,..., an} C A of nonzero elements with Vfc=i "/.■ = 1 and a,k A ai = 0 for k # I. Denote by V := V(A) the collection of all finite partitions in A. Take pi,p2 E V. We say that pi is finer than p2 if a = \/{b E p± \ b < a} for each a E The idealization principle (in Nelson's credo) or the saturation principle (in Robinson's credo) guarantees that in every standard Boolean algebra we may find a hyperfinite partition which is finer than each standard finite partition. We call this object an infinitely fine partition.
V. Troitskii noticed that an infinitely fine partition possesses some additional properties that can be used for a nonstandard approach to atomic decomposition.
3.1. Hammer^Sobczik Decomposition Theorem. Let p, be a finite finitely additive positive measure on a Boolean algebra A. Then there exist a sequence (Sn)n£n of distinct zero-one measures on A, a sequence (rn)n€N of positive reals, and a strongly continuous positive measure p0 on A, such that rn < 00
and ¡j, = p0 + J2™=1rnSn. Furthermore, this decomposition is unique.
< Let V be an infinitely fine partition of A. Take as pi an element of V of greatest measure. Let r\ = 0p{pi) and ¿i = SPl. If p is strongly continuous then ri = 0. It follows that pi = p — riSi is a positive standard measure. Now we may apply this procedure to pi and obtain p2 = p± — ^62, etc. Iterating the process, we obtain the decreasing sequence (rn)neoN of standard positive reals, the sequence (5n)ne°N of standard distinct zero-one measures. By the transfer principle we may extend these sequences to some standard sequences indexed by N. It is easy that po defined by //(, = p ^ Yl^Li rn$n is a standard strongly continuous positive measure. >
The Boolean-valued approach deals with vector-valued measures. We introduce a new notion of atomicity, and prove a corresponding result on atomic decomposition. As above, we denote by ®:= ^i(F) the Boolean algebra of all band projections in F.
A positive measure p E ba (A, F) is ®-discrete if for every measure v E ba (A, F) such that 0 < v < p the representation v = ap holds for some operator a : F F satisfying 0 < a < If-
We say that a measure n is ®-atomic if n belongs to the band in ba (A, F) generated by B-discrete elements, and ¡j, is M-diffuse if ¡j, belongs to the complementary band.
The essential starting point for B-atomic decomposition is the following result on Boolean-valued representation of vector measures, see [12].
3.2. Theorem. For every measure // e ba M. /•') there exists a unique element m E such that [m e ba (A, 71)} = 1 and ¡m(a) = m(a)] = 1 a E A. The correspondence («4 m is a lattice isomorphism from ba (A, F) to ba (A, TZ)^. If F = Til then the isomorphism is a bijection. Moreover,
(1) {j, is M-discrete if and only if m is discrete inside ;
(2) ¡j, is M-atomic if and only if m is atomic inside V®;
(3) fi is M-diffuse if and only if m is diffuse inside V'^.
We are now in a position to state the B-atomic decomposition theorem and the fact that the B-atomic fragment of a vector measure is the sum of a disjoint sequence of «spectral measures,» see [13].
3.3. Theorem. Let ¡j, : A F be a positive bounded measure. Then there exist a positive M-diffuse measure //(, : A —F. a decreasing sequence of positive elements (Fn)n£N in F, and a sequence ofpairwise disjoint Boolean homomorphisms hn: A—y ®, riGN, such that the series fn -is order convergent and
oo
li{a) = /i0(a) + XI hn(a)fn (a E A).
n=1
The representation of ¡j, in this form is unique.
The problem of extending a finitely additive vector measure may be treated analogously by successive application of the two nonstandard approaches we discuss, see [14].
4. Boolean-valued Banach spaces
Here we expose the Boolean-valued approach to studying Banach spaces with complete Boolean algebras of projections. These spaces appear in various branches of analysis, most frequently in the theory of operator algebras [4, 15, 16].
Let X be a Banach space and let C(X) stand for the set of all bounded linear operators on X. Assume that a mapping (p : B —y £(X) is injective and satisfies the following conditions:
(1) ip(b) is a projection with norm one for all b E B; moreover, ip( 1) = Ix and ¥>(0) = 0;
(2) the projections ip(b) and <p(b') commute for arbitrary 6, b' E B;
(3) the equalities ip(b V b') = ip(b) o ip(b') and <p(b*) = Ix — b) hold for all b and b'.
In this case the set B := </?(B) is referred to as & complete Boolean algebra of projections in the space X. The above situation is symbolized by ® IZ C(X) and the Boolean algebras ® and B are identified. If (e^gs is a partition of unity in ® and is a family in X then an element x E X for which e^x^ = e^x (£ E S)
is called the mixing of (x^) given (e^). A Banach space X is M-cyclic if B C C(X) and the following conditions hold:
(4) the mixing of an arbitrary bounded family in X given a partition of unity in ® (with the same index set) exists and is unique;
(5) the unit ball of X is closed under mixing.
The simplest example of a B-cyclic Banach space is the space of continuous functions on the Stone space of B. Clearly, an .HI''-algebra and an JW-algebra are B-cyclic Banach spaces.
Let X and Y be Banach spaces; moreover, assume that B C C(X) and B C C(Y). An operator T : X —y Y is B-linear if T is linear and commutes with projections in B. The set of all bounded B-linear operators from X into Y is B-cyclic if Y is a B-cyclic Banach space. We call a bijective B-linear operator a B-isomorphism; if, in addition, the operator is norm-preserving, we speak about an isometric B-isomorphism.
If X is a Banach space inside then the restricted descent of X consists of elements x E X-l satisfying [||a;|| < c] = 1 for some cel.
4.1. Theorem. The restricted descent of a Banach space within V® is a M-cyclic Banach space. Conversely if X is a M-cyclic Banach space, then there is a Banach space X in unique up to isometric isomorphism, such that the restricted descent of X is isometrically M-isomorphic to X.
The notions of nonstandard hull and p-nonarehimedean hull of a normed space are very useful patent tools of the geometric theory of Banach spaces. These notions were originally introduced by W. A. J. Luxemburg as a development of ideas of A. Robinson. It seems interesting to extend both notions to B-cyclic Banach spaces. This could provide a unified view of many results on the nonstandard hulls of Banach spaces and Banach lattices.
4.2. Problem. Develop a theory of nonstandard hulls in a Boolean-valued model. Study the corresponding construction of the «descended» nonstandard hull.
To address the problem needs developing and simplifying the formal facilities that appear along the described path of combining descents and ascents with an intermediate application of Robinson's infinitesimal analysis.
In closing, we observe that there are at least three possibilities of combining nonstandard methods: the first consists in implementing infinitesimal constructions in a Boolean-valued model; the second, in applying nonstandard methods successively; and the third, in seeking for Boolean-valued interpretations of internal and external set theories. Each of these approaches involves its own technical advantages
and disadvantages and should be judged upon its own merits and demerits.
References
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г. Владикавказ
Статья поступила 9 февраля 2000 г.
