RUSSIANACADEMYOFSCIENCES MINISTRYOFEDUCATIONANDSCIENCE OFTHERUSSIANFEDERATION VLADIKAVKAZSCIENTIFICCENTER SOUTHERNMATHEMATICALINSTITUTE NORTHOSSETIANSTATEUNIVERSITY TRENDS IN SCIENCE • THE SOUTH OF RUSSIA A MATHEMATICAL MONOGRAPH Issue 6 BOOLEAN VALUED ANALYSIS: SELECTED TOPICS by A. G. Kusraev and S. S. Kutateladze Vladikavkaz 2014 ББК 22.16 УДК517.98 K94 Editor A.E. Gutman Reviewers: S.A. Malyugin and E.Yu. Emelyanov Editors of theseries: Yu.F. Korobeinik and A.G. Kusraev Kusraev A. G. and Kutateladze S. S. Boolean Valued Analysis: Selected Topics / Ed. A.E. Gutman.—Vladi- kavkaz: SMI VSC RAS, 2014.—iv+400 p.—(Trends in Science: The South of Russia. A Mathematical Monograph. Issue6). The book treats Boolean valued analysis. This term signifies the technique of studying properties of an arbitrary mathematical object by means of comparison between its representations intwo different set-theoretic models whose construction utilizes principally distinct Boolean algebras. As these models, we usually take the classicalCantorianparadiseintheshapeofthevonNeumannuniverseandaspecially- trimmedBooleanvalueduniverseinwhichtheconventionalset-theoreticconceptsand propositions acquire bizarre interpretations. Exposition focuses on the fundamental properties oforderboundedoperators invectorlattices.Thisvolumeisintendedfor theclassicalanalystseekingnewpowerful toolsandforthemodeltheoristinsearch ofchallengingapplicationsofnonstandard modelsofsettheory. Кусраев А. Г., Кутателадзе С. С. Булевозначный анализ: Избранные темы / отв. ред. А. Е. Гутман.— Владикавказ:ЮМИ ВНЦ РАНиРСО-А,2014.—iv+400 с.—(Итоги науки. Юг России. Математическая монография. Вып. 6). Монографияпосвященабулевозначномуанализу.Такназываютаппаратис- следованияпроизвольныхматематическихобъектов,основанныйнасравнитель- ном изучении их вида в двух моделях теории множеств, конструкции которых основаны на принципиально различных булевых алгебрах. В качестве этих мо- делейфигурируютклассическийкантороврайвформеуниверсумафонНейма- на и специально построенный булевозначный универсум, в котором теоретико- множественные понятия и утверждения получают весьма нетрадиционные тол- кования. Основное внимание уделено фундаментальным свойствам порядково ограниченных операторов в векторных решетках. Книга ориентирована на ши- рокий круг читателей, интересующихся современными теоретико-модельными методамивихприложениикфункциональному анализу. ISBN 978-5-904695-24-8 (cid:2)c Southern MathematicalInstitute VSCRAS&RNO-A,2014 (cid:2)c A.G.Kusraev,2014 (cid:2)c S.S.Kutateladze, 2014 PREFACE Humans definitely feel truth but cannot define truth properly. That iswhatAlfredTarskiexplainedtousinthe 1930s.Mathematicspursues truthbywayofproof,aswittilyphrasedbySaundersMacLane.Boolean valued analysis is one of the vehicles of the pursuit, resulting from the fusion of analysis and model theory. Analysis is the technique of differentiation and integration. Diffe- rentiation discovers trends, and integration forecasts the future from trends. Analysis opens ways to understanding of the universe. Model theory evaluates and counts truth and proof. The chase of truth not only leads us close to the truth we pursue but also enables us to nearly catch up with many other instances of truth which we were not aware nor even foresaw at the start of the rally pursuit. That is what we have learned from Boolean valued models of set theory. These models stem from the famous works by Paul Cohen on the continuum hypothesis. They belong to logic and yield a profusion of the surprising and unforeseen visualizations of the ingredients of mathematics. Many promising opportunities are open to modeling the powerful habits of reasoning and verification. Logicorganizesandordersourwaysofthinking,manumittingusfrom conservatism in choosing the objects and methods of research. Logic of today is a fine instrument of pursuing truth and an indispensable insti- tution ofmathematicalfreedom.Logicliberates mathematics,providing nonstandard ways of reasoning. Some modelof settheory is nonstandard if the membership between the objects of the model differs from that of the originals. In fact, the nonstandardtools oftodayuse a couple ofset-theoreticmodels simulta- neously. Boolean valued models reside within the most popular logical tools. Boolean valued analysis is a blending of analysis and Boolean val- ued models which originated and distinguishes itself by ascending and descending, mixing, cycling hulls, etc. iv Preface In this book we show how Boolean valued analysis transforms the theory of operators in vector lattices. We focus on the recent results that were not reflected in the monographic literature yet. InChapter1wecollecttheBooleanvaluedprerequisitesofthefurther analysis.Chapter2providesthe presentationoftherealsandcomplexes within Booleanvaluedmodels.In Chapter3 we givethe Booleanvalued interpretations of order bounded operators with the emphasis on lattice homomorphisms and disjointness preserving operators. Chapter 4 con- tains the solutionofthe Wicksteadproblemas wellas other new results onbandpreservingoperators.Chapter 5 dealswith variousapplications ofordercontinuousoperatorsto injective Banachlattices,Maharamop- erators, and related topics. Adaptationofthe ideasofBooleanvaluedmodelstofunctionalanal- ysis projects among the most important directions of developing the synthetic methods of mathematics. This approach yields the new mod- els of numbers, spaces, and types of equations. The content expands of allavailabletheorems andalgorithms.The whole methodologyof math- ematicalresearchisenrichedandrenewed,openingupabsolutelyfantas- tic opportunities. We can now transform matrices into numbers, embed function spaces into a straight line, yet having still uncharted vast ter- ritories of new knowledge. The chase of truth not only leads us close to the truth we pursue but also enables us to nearly catch up with many other instances of truth which we were unaware nor even foresaw at the start of the rally pursuit. That is what we have learned from Boolean valued analysis. Quite a long time had passed until the classical functional analysis occupieditspresentpositionofthelanguageofcontinuousmathematics. Nowthetimehascomeofthenewpowerfultechnologiesofmodeltheory inmathematicalanalysis.Notalltheoreticalandappliedmathematicians havealreadygainedthe importance ofmoderntools andlearnedhow to use them. However,there is no backwardtraffic in science, and the new methods are doomed to reside in the realmof mathematics for ever and they will shortly become as elementary and omnipresent in analysis as Banach spaces and linear operators. A. Kusraev S. Kutateladze CHAPTER 1 BOOLEAN VALUED REQUISITES In this chapter we briefly present some prerequisites of the theory of Boolean valued models. All missing details may be found in Bell [43], Jech[184],KusraevandKutateladze[248,249],TakeutiandZaring[388]. We mainly keep the notation of [248] and [249]. The most important feature of Boolean valued analysis consists in comparativeanalysisofthe standardandnonstandard(Booleanvalued) models under consideration which uses the special technique of ascend- ing and descending. Moreover, it is often necessary to carry out some syntactic comparison of formal texts. Therefore, before we launch into theascendinganddescendingmachinery,wehavetograspacleareridea of the status of mathematical objects in the framework of a formal set theory, the construction of a Boolean valued universe, and the way of assigningtheBooleantruthvaluetoeachsentenceofthelanguageofset theory. We use several notations for implication: ⇒ presents the Boolean operation,=⇒ stands for the logicalconnective, but often in a set theo- reticformulaweuse→insteadof=⇒indicatingthatthisformulawillbe interpretedinsome Booleanvaluedmodel. We alsouse ↔,⇔, and⇐⇒ with the similar meaning. The proof-theoretic consequence relation (cid:7) is applied alongside with the semantic (or model-theoretic) consequence relation |=. Observethat,speakingofaformalsettheory,wewillfreely(because thisisinfactunavoidable)adheretothelevelofrigorwhichiscurrentin the mainstream of mathematics and introduce abbreviations by means of the definor, i.e. the assignment operator, := without specifying any subtleties. 2 Chapter 1. Boolean Valued Requisites 1.1. Zermelo–Fraenkel Set Theory At present, the most widespread axiomatic foundation for mathe- matics is Zermelo–Fraenkel set theory. We will briefly recall some of its concepts, outlining the details we need in the sequel. 1.1.1. The alphabet of Zermelo–Fraenkel theory ZF or ZFC, if the presence of choice AC is stressed, comprises the symbols of variables; the parentheses ( and ); the propositional connectives (i.e., the signs of propositional calculus) ∨, ∧, →, ↔, and ¬; the quantifiers ∀ and ∃; the equality sign =; and the symbol of the special binary predicate of containment or membership ∈. In general, the domain of the variables of ZF is thought as the world or universe of sets. In other words, the universe of ZF contains nothing but sets. We write x ∈ y rather than ∈(x,y) and say that x is an element of y. 1.1.2. The formulas of ZF are defined by the routine procedure. In otherwords,theformulasofZFarefinitetextsresultingfromtheatomic formulasx=y and x∈y, where x andy arevariablesof ZF,by reason- ably placing parentheses, propositional connectives, and quantifiers ϕ∨ψ, ϕ∧ψ, ¬ϕ, ϕ→ψ, ϕ↔ψ, (∀x)ϕ, (∃x)ϕ. So, if ϕ1 and ϕ2 are formulas of ZF and x is a variable then the texts ϕ1 → ϕ2 and (∃x)(ϕ1 → (∀y)ϕ2)∨ϕ1 are formulas of ZF, whereas ϕ1∃x and ∀(x∃ϕ1¬ϕ2 are not. We attach the natural meaning to the termsfreeandbound variables andthe termdomain of a quantifier. For instance, in the formula (∀x)(x ∈ y) the variable x is bound and the variable y is free, whereas in the formula (∃y)(x = y) the variable x is free and y is bound (for it is bounded by a quantifier). Henceforth, in order to emphasize that the only free variables in a formula ϕ are the variables x1,...,xn, we write ϕ(x1,...,xn). Sometimes such a formula is considered as a “function”; in this event, it is convenient to write ϕ(·,...,·)orϕ=ϕ(x1,...,xn),implying thatϕ(y1,...,yn)isaformula of ZF obtained by replacing each free occurrence of x by y for k:= k k 1,...,n. 1.1.3. Studying ZF, it is convenient to use some expressive tools absent in the formal language. In particular, in the sequel it is worth- while employing the concepts of class and definable class and also the corresponding symbols of classifiers like Aϕ:= Aϕ(·):= {x : ϕ(x)} and Aψ:=Aψ(·,y):={x:ψ(x,y)},whereϕandψareformulasofZFandy is 1.1. Zermelo–Fraenkel Set Theory 3 adistinguishedcollectionofvariables. Ifitisdesirabletoclarifyorelim- inate the appearing records then we can assume that the use of classes and classifiers is connected only with the conventionalagreement on in- troducing abbreviations. This agreement, sometimes called the Church schema, reads: z ∈{x:ϕ(x)}↔ϕ(z), z ∈{x:ψ(x,y)}↔ψ(z,y). 1.1.4.WorkingwithinZF,wewillusesomenotationsthatarewidely spread in mathematics. We start with the most frequent abbreviations: x(cid:13)=y:=¬x=y, x∈/ y:=¬x∈y; (∀x∈y)ϕ(x):=(∀x)(x∈y →ϕ(x)); (∃!z)ϕ(z):=(∃z)ϕ(z)∧((∀x)(∀y)(ϕ(x)∧ϕ(y)→x=y)); (∃x∈y)ϕ(x):=(∃x)(x∈y∧ϕ(x)). Theempty set∅,the pair{x,y},the singleton{x},the orderedpair (x,y), and the ordered n-tuple (x1,...,xn) are defined as ∅:={x: x(cid:13)=x}; {x,y}:={z : z =x∨z =y}, {x}:={x,x}, (x,y):={x,{x,y}}; (x1,...,xn):=((x1,...,xn−1),xn). (cid:2) (cid:3) Theinclusion⊂,theunion ,theintersection ,thepowersetP(·), and the universe of sets V are introduced as follows: x⊂y:=(∀z)(z ∈x→z ∈y); (cid:4) x:={z :(∃y ∈x)z ∈y}; (cid:5) x:={z :(∀y ∈x)z ∈y}; P(x):=“the class of all subsets of x”:={z : z ⊂x}; V:=“the class of all sets”:={x: x=x}. 4 Chapter 1. Boolean Valued Requisites Notealsothatinthesequelweacceptmorecomplicateddescriptions in which much is presumed: Fnc(f):=“f is a function”; dom(f):=“the domain of f”; im(f):=“the range of f”; ϕ(cid:7)ψ:=ϕ→ψ:=“ψ is derivable from ϕ”; “a class A is a set”:=A∈V:=(∃x)(∀y)(y ∈A↔y ∈x). Similar simplifications will be used in rendering more complicated for- mulas without further stipulation. For instance, instead of some rather involved formulas of ZF we simply write f :x→y ≡“f is a function from x to y”; “X is a vector lattice”; U ∈L∼(X,Y)≡“U is an order bounded linear operator from X to Y.” 1.1.5. In ZFC, we accept the usual axioms and rules of a first-order theorywithequalitywhichfixthestandardmeansofclassicalreasoning. Recall the equality axioms: (1) (∀x)x=x (reflexivity); (2) (∀x)(∀x)x=y →y =x (symmetry); (3) (∀x)(∀y)(∀y)x=y∧y =x→x=z (transitivity); (cid:6) (4) (∀x)(∀y)(∀u)(∀v) (x=y∧u=v) (cid:7) →(x∈u→y ∈v) (substitution). 1.1.6. The classical first-order logic CL has the following axiom schemas (ϕ, ψ, and ω are arbitrary formulas of CL): (1) ϕ→(ϕ∧ϕ); (2) (ϕ∧ψ)→(ψ∧ϕ); (3) (ϕ→ψ)→((ϕ∧ω)→(ψ∧ω)); (4) ((ϕ→ψ)∧(ψ →ω))→(ϕ→ω); (5) ψ →(ϕ→ψ); (6) (ϕ∧(ϕ→ψ))→ψ; (7) ϕ→(ϕ∨ψ); (8) (ϕ∨ψ)→(ψ∨ϕ); 1.1. Zermelo–Fraenkel Set Theory 5 (9) ((ϕ→ω)∧(ψ →ω))→((ϕ∧ψ)→ω); (10) ¬ϕ→(ϕ→ψ); (11) ((ϕ→ψ)∧(ϕ→¬ψ))→¬ϕ; (12) ϕ∨¬ϕ. The only rule of inference in CL is modus ponens: (MP) If ϕ and ϕ→ψ are provable in CL then so is ψ. 1.1.7.Moreover,the followingspecial orproper axiomsareaccepted in ZFC as a correct formalization of the principles of most mathemati- cians working with sets: (1)AxiomofExtensionality. If x and y have the same elements then x=y: (∀x)(∀y)(x⊂y∧y ⊂x→x=y). (cid:2) (2) Axiom of Union. To each x there exists a set y = x: (cid:8) (cid:4) (cid:9) (∀x)(∃y) y = x . (3) Axiom of Powerset. To each x there exists a set y =P(x): (∀x)(∃y)(y =P(x)). (4) Axiom Schema of Replacement. If a class A is a function then ϕ to each x there exists a set v =A (x)={A (z): z ∈x}: ϕ ϕ (∀x)((∀y)(∀z)(∀u)ϕ(y,z)∧ϕ(y,u)→z =u) →(∃v)(v ={z :(∃y ∈x)ϕ(y,z)}). (5) Axiom of Foundation. Each nonempty set has an ∈-minimal element: (∀x)(x(cid:13)=∅→(∃y ∈x)(y∩x=∅)). (6) Axiom of Infinity. There exists an inductive set: (∃ω)(∅∈ω)∧(∀x∈ω)(x∪{x}∈ω). (7) Axiom of Choice. Each family of nonempty sets has a choice function: (∀F)(∀x)(∀y)((x(cid:13)=∅∧F :x→P(y)) →((∃f)f :x→y∧(∀z ∈x)f(z)∈F(z)).