Dominated operators

Contents

Foreword to the English Translation

ix

Preface

xi

Chapter 1. Boolean Algebras and Vector Lattices § 1.1. Boolean Algebras

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§ 1.2. Representation of Boolean Algebras § 1.3. Vector Lattices

10

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17

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25

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33

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40

§ 1.5. Normed Vector Lattices

Chapter 2. Lattice-Normed Spaces § 2.1. Preliminaries § 2.2. Completion § 2.3. Examples

2

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§ 1.4. Representation of Vector Lattices

§ 1.6. Comments

1

44

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45

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54

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62

§ 2.4. Continuous Banach Bundles

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70

§ 2.5. Measurable Banach Bundles

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79

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84

§ 2.6. Comments

vi

Contents

Chapter 3. Positive Operators

89

§ 3.1. Operators in Vector Lattices

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§ 3.2. Fragments of a Positive Operator

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§ 3.3. Orthomorphisms and Lattice Homomorphisms § 3.4. Maharam Operators

99

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108

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118

§ 3.5. Maharam’s Extension of Positive Operators § 3.6. Comments

90

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126

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133

Chapter 4. Dominated Operators

141

§ 4.1. The Space of Dominated Operators

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§ 4.2. Decomposability of the Space of Dominated Operators § 4.3. Order Continuous Operators

142

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150

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156

§ 4.4. The Yosida–Hewitt-Type Theorems

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163

§ 4.5. Extension of Dominated Operators

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171

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179

§ 4.6. Comments

Chapter 5. Disjointness Preserving Operators § 5.1. Band Preserving Operators § 5.2. n-Disjoint Operators

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188

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195

§ 5.3. Weight-Shift-Weight Factorization

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206

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214

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221

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230

§ 5.4. Multiplicative Representation § 5.5. Decomposable Operators § 5.6. Comments

187

Chapter 6. Integral Operators § 6.1. Vector Integration

236

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237

§ 6.2. Integral Representation by Quasi-Radon Measures

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248

§ 6.3. Functional Representation of Maharam’s Extension

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257

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266

§ 6.4. Integral Operators

§ 6.5. Pseudointegral Operators § 6.6. Comments

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278

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286

Contents

vii

Chapter 7. Operators in Spaces with Mixed Norm § 7.1. Spaces with Mixed Norm § 7.2. Summing Operators

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292

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299

§ 7.3. Isometric Classification

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§ 7.4. Kaplansky–Hilbert Modules

316

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325

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334

Chapter 8. Applications of Boolean-Valued Analysis § 8.1. Real Numbers in Boolean-Valued Models

340

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352

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362

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371

§ 8.3. Boolean-Valued Banach Spaces § 8.4. Involutive Banach Algebras

338

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§ 8.2. Boolean-Valued Analysis of Vector Lattices

§ 8.5. Cyclically Compact Operators § 8.6. Comments

307

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§ 7.5. AW ∗ -Algebras § 7.4. Comments

291

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379

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389

Appendix. Boolean-Valued Models

394

References

412

Symbol Index

434

Subject Index

437

Foreword to the English Translation

The topic of this book belongs to vector lattice theory. Dominated operators are remote relatives of the noble family of bounded linear operators and functionals residing in Banach space. The concept of a dominated operator was invented in the 1930s by Leonid Vital 0 evich Kantorovich, a renowned mathematician and a Nobel prize winner in economics. This concept utilizes the main sociological trick of functional analysis which rests on studying the population of some mathematical objects in order to reveal their individual features and acquired traits. The theory of dominated operators has ripened in the recent decades mainly due to the contribution by Professor A. G. Kusraev and his students in Vladikavkaz and Novosibirsk. A few years ago Professor A. G. Kusraev asked me to edit and introduce the English translation of his book. I undertook the task readily for two reasons: First, the topic of the book is attractive and close to my own area of research. Second, I feel proud of the achievements of Professor A. G. Kusraev, once a brilliant student of mine at Novosibirsk State University and now my inseparable fellow with whom I have been sharing many splendid days full of inspirational mathematics. Unfortunately, the Russian version of this book is still in preparation in view of the unstable situation of the Northern Caucasus which hindered us in working on the translation. I hope that the reader will take this regrettable circumstance into account and forgive the inadvertent shortcomings of the inadequate communication between the author and the editor. S. Kutateladze Novosibirsk Akademgorodok

Preface

The notion of a dominated or majorized operator rests on a simple idea that goes as far back as the Cauchy method of majorants. Loosely speaking, the idea can be expressed as follows. If an operator (equation) under study is dominated by another operator (equation), called a dominant or majorant, then the properties of the latter have a substantial influence on the properties of the former. Thus, operators or equations that have “nice” dominants must possess “nice” properties. In other words, an operator with a somehow qualified dominant must be qualified itself. Mathematical tools, putting the idea of domination into a natural and complete form, were suggested by L. V. Kantorovich in 1935–36. He introduced the fundamental notion of a vector space normed by elements of a vector lattice and that of a linear operator between such spaces which is dominated by a positive linear or monotone sublinear operator. He also applied these notions to solving functional equations. In the succeeding years many authors studied various particular cases of latticenormed spaces and different classes of dominated operators. However, research was performed within and in the spirit of the theory of vector and normed lattices. So, it is not an exaggeration to say that dominated operators, as independent objects of investigation, were beyond the reach of specialists for half a century. As a consequence, the most important structural properties and some interesting applications of dominated operators have become available since recently. By the early 1980s, certain qualitative changes took place in the theory of vector lattices. New methods were suggested, while the range of applications was considerably extended and enriched. Radically new ideas came from other branches of mathematics. All these facts provided necessary prerequisites for a deep study of dominated operators and led to formation of a novel theory of dominated operators. The aim of this book is to present the main results on dominated operators which were obtained in the last fifteen years, thus demonstrating a certain ripeness of the theory. The book consists of eight chapters. Chapter 1 contains definitions and prelim-

xii

Preface

inary information about Boolean algebras and vector lattices. Chapter 1 is included mainly for fixing terminology and notation. In Chapter 2 some structural properties are considered of a vector space endowed with a norm taking values in some vector lattice. Here, we address the questions of completing these spaces as well as representing them by means of continuous Banach bundles and measurable Banach bundles. Chapter 3 is devoted to positive operators. The content of Chapter 3 is traditional except for some new results concerning fragments and order-intervalpreserving extension of positive operators. In Chapter 4 we study the general properties of dominated operators. A dominated operator has the least (or exact) dominant under rather weak assumptions. Assigning to each dominant operator its exact dominant we come to some vector norm with values in the vector lattice of regular operators. The central result of this chapter is decomposability of this lattice-normed space of dominants. In Chapter 5 disjointness preserving and decomposable operators are considered. In particular, we give their analytic representations and decompositions into simpler parts. Chapter 6 deals with integrality and pseudointegrality for dominated operators. It turns out that a dominated operator inherits these properties from every dominant. From this fact we deduce criteria for weak integrality and pseudointegrality of dominated operators. Several assertions about the general form of dominated operators are also stated. Various classes of operators under study in functional analysis are often defined using the terms that mix norms and orders. This theme is developed in Chapter 7 in which we introduce some new classes of spaces and operators. The problem of isometric classification of spaces with mixed norm is also briefly discussed. Chapter 8 is devoted to the so-called Boolean-valued analysis of vector lattices and dominated operators. Our starting point is the assertion claiming that each field of the reals in a Boolean-valued model gives rise to a universally complete vector lattice. Therefore, a huge part of the general theory of vector lattices admits a straightforward derivation by translating and interpreting the well known properties of the conventional reals. This chapter also exposes the Boolean-valued approach to more advanced sections of analysis such as lattice-normed spaces, involutive Banach algebras, etc. Elementary exposition of the apparatus of Booleanvalued model theory resides in the Appendix. The diversity of topics and results we handle in the book determines the style of exposition. Information freely accessible to the reader is given without demonstration. All principal results are however furnished with complete proofs. Comments to all chapters contain some additional remarks and a guide to the literature. While writing the book, the author assumed the reader familiar with the standard courses

Preface

xiii

in the theory of vector lattices and positive operators. I seize the opportunity to express my gratitude to all those who helped me in preparation of the book. My pleasant debt is to acknowledge the financial support of the Sobolev Institute of Mathematics of the Siberian Division of the Russian Academy of Sciences, the North Ossetian Scientific Center, the North Ossetian State University, the Russian Foundation for Basic Research, the International Science Foundation, and the American Mathematical Society during the compilation of the monograph. July, 1999

A. G. Kusraev

Chapter 1 Boolean Algebras and Vector Lattices

The present chapter collects some basic facts from the theories of Boolean algebras and vector lattices. It lays the foundation for the theory of lattice-normed spaces and dominated operators to which the book is devoted. An elementary algebraic theory of Boolean algebras is briefly presented in Section 1.1. We expose the main constructions with Boolean algebras such as subalgebras, homomorphic images, ideals, factor algebras, Cartesian products, and partitions of unity (1.1.5). We state a very useful Exhaustion Principle for Boolean algebras and a pair of its corollaries (1.1.6). Some important examples of Boolean algebras are also provided: the algebras of regular open sets and clopen sets, the algebras of Borel sets modulo meager sets and measurable sets modulo negligible sets, etc. (1.1.7). Three representation theorems are proved in Section 1.2: the celebrated Stone Representation Theorem (1.2.3), the Loomis–Sikorski Theorem on representation of σ-complete Boolean algebras (1.2.6), and representation of multinormed Boolean algebras by measure spaces with direct sum property (1.2.10). Ogasawara’s characterization of order completeness of Boolean algebras (1.2.4) and the Sikorski representation of a Boolean homomorphism (1.2.5) via the Stone space are given. The concept of lifting and the corresponding immersion of a measure space into the Stone space of the respective complete Boolean algebra are also presented (1.2.7, 1.2.8). Section 1.3 begins with the notions of vector lattice and vector sublattice, ideal and factor lattice, order and relative uniform convergence, etc. (1.3.2, 1.3.4– 1.3.6). The disjointness relation in a vector lattice provides the three basic Boolean algebras of bands, band projections, and fragments of an order-unity (1.3.3). These are isomorphic to one another for an order complete vector lattice with order-unity (1.3.7). The spectral function of an element of vector lattice and a list of its useful properties are given (1.3.8). The Freudenthal Spectral Theorem is stated as well as a characterization of general vector lattices in which the weak and strong forms of this theorem hold (1.3.9).

2

Chapter 1

Three basic examples of universally complete vector lattices and some interconnections between them, as well as the representation of an arbitrary order complete vector lattice as an order-dense ideal in such spaces is the content of Section 1.4. The first example is the space of continuous functions on a quasiextremal compact space which assume possibly infinite values on a nowhere-dense set depending on a function (1.4.1, 1.4.2). This space is an order σ-complete vector lattice and even an order complete vector lattice provided that the underlying compact space is extremally disconnected (1.4.2). The second example is the space of all spectral functions with values in a σ-complete Boolean algebra (1.4.3). The Stone transform of a σ-complete Boolean algebra can be extended to a linear and lattice isomorphism of the corresponding space of spectral functions onto the space of extended valued continuous functions on the Stone space (1.4.4). Using the Freudenthal Spectral Theorem, we prove that an arbitrary order complete vector lattice is isomorphic to an order-dense ideal of the space of extended valued continuous functions on an extremal compact space (1.4.5, 1.4.6). The third example is the space of (equivalence classes of) measurable functions which is order complete provided that the underlying measure space has the direct sum property (1.4.7). In this case the space of essentially bounded measurable functions admits lifting (1.4.8); moreover, each lifting generates a linear and lattice isomorphism between the space of measurable functions and the space of extended valued continuous functions (1.4.9). Various conditions under which an abstract order complete vector space is isomorphic to an order-dense ideal in some space of measurable functions are also presented (1.4.10). Section 1.5 starts with basic definitions and elementary facts of the theory of normed vector lattices (1.5.1, 1.5.2). We then discuss various useful characterizations of Banach lattices with order continuous norm (1.5.3), as well as monotonically complete Banach lattices and Banach lattices with order semicontinuous norm (1.5.4). Abstract M - and Lp -spaces are introduced (1.5.5) and the corresponding representation theorems are also stated (1.5.6). In particular, every Banach lattice has some AM -structure locally. This observation opens a way to some functional calculus in an arbitrary Banach lattice (1.5.7). Two more applications of a local AM -structure are stated: a disjointly complete Banach lattice has the projection property (1.5.10) and a Banach lattice is order complete if and only if it is disjointly complete (1.5.11). To prove these results, some order properties (1.5.8, 1.5.9) of an AM -space with unity are needed. 1.1. Boolean Algebras In this section we sketch a minimum about Boolean algebras which we need in the sequel. A more explicit exposition may be found elsewhere; for instance, cf. [127, 283, 352, 384].

Boolean Algebras and Vector Lattices

3

1.1.1. To fix terminology and notation, we recall some well-known notions. An ordered set is a pair (M, ≤), where ≤ is an order on M (see A.1.10). An ordered set is also called a partially ordered set or, briefly, a poset. It is in common parlance to apply all names of (M, ≤) to the underlying set M of (M, ≤). We indulge in doing the same elsewhere without further circumlocution. An upper bound of a subset X of a poset M is an element a ∈ M such that x ≤ a for all x ∈ X. A least element of the set of upper bounds of X is called a least upper bound or supremum of X and denoted by sup(X) or sup X. In other words, a = sup(X) if and only if a is an upper bound of X and a ≤ b for every upper bound b of X. By reversal, i.e., by passing from the original order ≤ on a poset M to the reverse or opposite order ≤−1 (x ≤−1 ⇔ y ≤ x), define a lower bound of a subset X of M and a greatest lower bound, inf(X) of X, also called an infimum of X and denoted inf X. If a least upper or greatest lower bound of a set in M exists then it is unique and so deserves the definite article. It can be easily checked that the following commutativity and associativity laws hold in every poset on duly stipulating existence of the suprema and infima in question: (1) sup sup xα,β = sup sup xα,β ; α∈A β∈B

β∈B α∈A

(2) inf inf xα,β = inf inf xα,β ; α∈A β∈B β∈B α∈A
S (3) sup Xα = sup sup Xα ; α∈A α∈A
S Xα = inf inf Xα . (4) inf α∈A

α∈A

1.1.2. A lattice is an ordered set L in which each pair {x, y} has the join x ∨ y := sup{x, y} and meet x ∧ y := inf{x, y}. Given a subset X of a lattice L, we use the notation: _ ^ X := sup(X), X := inf(X), _ _ ^ ^ xα := {xα : α ∈ A}, xα := {xα : α ∈ A}, α∈A

α∈A n _ k=1 n ^

xk := x1 ∨ · · · ∨ xn := sup{x1 , . . . , xn }, xk := x1 ∧ · · · ∧ xn := inf{x1 , . . . , xn }.

k=1

Here (xα )α∈A is a family in L, and x1 , . . . , xn stand for some members of L. The binary operations join (x, y) 7→ x ∨ y and meet (x, y) 7→ x ∧ y act in every lattice L and possess the following properties:

4

Chapter 1 (1) commutativity: x ∨ y = y ∨ x,

x ∧ y = y ∧ x;

(2) associativity: x ∨ (y ∨ z) = (x ∨ y) ∨ z,

x ∧ (y ∧ z) = (x ∧ y) ∧ z.

By induction, from (2) we deduce that each nonempty finite set in a lattice has the meet and join. If each subset of a lattice L has the supremum and infimum then L is a complete lattice. A lattice L is distributive provided that the following distributive laws are valid: (3) x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z); (4) x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). If a lattice L has the least or greatest element then the former is called the zero of L and the latter, the unity of L. The zero and unity of L are solemnly denoted by 0L and 1L . It is in common parlance to use the simpler symbols 0 and 1 and nicknames zero and unity provided that the context prompts the due details. Note also that 0 and 1 are neutral elements: (5) 0 ∨ x = x, 1 ∧ x = x. W V Specifying the general definitions, also note that ∅ = sup ∅ := 0 and ∅ = inf ∅ := 1. A complement x∗ of a member x of a lattice L with zero and unity is an element x∗ of L such that (6) x ∧ x∗ = 0, x ∨ x∗ = 1. Elements x and y in L are disjoint if x ∧ y = 0. So, every element x is disjoint from any complement x∗ . Note finally that if each element in L has at least one complement then we call L a complemented lattice. Recall by the way that a set U is disjoint whenever every two distinct members of U are disjoint. It is rather evident that an arbitrary lattice L may fail to have a complement to each element of L. 1.1.3. A Boolean algebra is a distributive complemented lattice with distinct zero and unity. Each element x of a Boolean algebra B has a unique complement denoted by x∗ . This gives rise to the mapping x 7→ x∗ (x ∈ B) which is idempotent (i.e., (∀ x ∈ B) (x∗∗ := (x∗ )∗ = x)) and presents a dual isomorphism or an antiisomorphism of B onto itself (i.e., it is an order isomorphism between (B, ≤) and (B, ≤−1 )). The three operations ∨, ∧, and ∗, living in every Boolean algebra B, are jointly referred to as Boolean operations. A Boolean algebra B is complete (σ-complete), if each subset (countable subset) of B has a supremum and an infimum. It is in common parlance to speak of σalgebras instead of σ-complete algebras.

Boolean Algebras and Vector Lattices

5

W V Associated with a Boolean algebra B, the mappings , : P(B) → B are available that ascribe to a set in B its supremum and infimum, respectively. These mappings are sometimes referred to as infinite operations. The infinite operations obey many important rules among which we for instance mention the De Morgan formulas  ∗  ∗ W V ∗ V W ∗ (1) xα = xα , xα = xα , α∈A

α∈A

α∈A

α∈A

and the infinite distributive laws V V (2) x ∨ xα = x ∨ xα , α∈A

(3) x ∧

W

α∈A

xα =

α∈A

W

x ∧ xα ,

α∈A

with xα ∈ B for all α ∈ A. 1.1.4. Let 2 := Z2 := P({∅}) := {0, 1} be the underlying set of the twoelement Boolean algebra now viewed as a field with the following operations: 0 + 0 := 0,

0 + 1 = 1 + 0 := 1,

0 · 1 = 1 · 0 := 0,

0 · 0 := 0,

1 + 1 := 0, 1 · 1 := 1.

Note that every member of 2 is idempotent. Consider an arbitrary set B with the structure of an associative ring whose every element is idempotent: (∀b ∈ B)(b2 = b). In this case B is called a Boolean ring. A Boolean ring is commutative and obeys the identity b = −b for b ∈ B. Each Boolean ring is obviously a vector space and, at the same time, a commutative algebra over 2. Recall that the unity of an algebra differs from its zero by definition. So, we may and will identify the field 2 with the subring of a Boolean ring comprising the zero and unity of the latter. We usually reflect the practice in symbols by letting 0 stand for the zero and 1, for the unity of whatever ring. This agreement leads clearly to a rather popular notational collision: the addition and multiplication of 2 may be redefined on making 0 play the role of 1 and vice versa. It is customary to endow a Boolean ring B with some order by the rule: b1 ≤ b2 ⇔ b1 b2 = b1

(b1 , b2 ∈ B).

The poset (B, ≤) obviously becomes a distributive lattice with the least element 0 and the greatest element 1. In this event, the lattice and ring operations are connected as follows: x ∨ y = x + y + xy,

x ∧ y = xy.

6

Chapter 1

Moreover, to each element b ∈ B there is a unique b∗ ∈ B, the complement of b, such that b∗ ∨ b = 1, b∗ ∧ b = 0. Obviously, b∗ = 1 + b. Hence, each Boolean ring is a Boolean algebra under the above order. In turn, we may transform a Boolean algebra B into a ring by putting x + y := x M y,

xy := x ∧ y

(x, y ∈ B),

where x M y := (x ∧ y ∗ ) ∨ (x∗ ∧ y) is a symmetric difference of x and y. In this case (B, +, · , 0, 1) becomes a unital Boolean ring whose natural order coincides with the initial order on B. Therefore, a Boolean algebra can be viewed as a unital algebra over 2 whose every element is idempotent. 1.1.5. Consider some methods of arranging new Boolean algebras. (1) A nonempty subset B0 of a Boolean algebra B is a subalgebra of B if B0 is closed under the Boolean operations ∨, ∧, and ∗; i.e., {x ∨ y, x ∧ y, x∗ } ⊂ B0 for all x, y ∈ B0 . Under the order induced from B, every subalgebra B0 is a Boolean algebra with the same zero and unity as those of B. In particular, B0 := {0B , 1B } is a subalgebra of B. A subalgebra B0 ⊂ B is regular or complete (σ-regularWor σ-complete) provided that for every set (countable set) A in B0 the elements A V and A, if exist in B, belong to B0 . The intersection of every family of subalgebras is itself a subalgebra. The same holds true for regular (σ-regular) subalgebras, which makes the definition to follow sound. The least subalgebra of B containing a nonempty subset M of B is the subalgebra generated by M . The regular (σ-regular) subalgebra generated by M is introduced in much the same manner. (2) An ideal of a Boolean algebra B is any nonempty set J in B obeying the conditions: x ∈ J, y ∈ J ⇒ x ∨ y ∈ J, x ∈ J, y ≤ x ⇒ y ∈ J. The set Ba := {x ∈ B : x ≤ a}, with a ∈ B, provides an example of an ideal of B. Such an ideal is called principal. If 0 6= e ∈ B then the principal ideal Be with the order induced from B is a Boolean algebra in its own right. The element e plays the role of unity in Be . The lattice operations of Be are inherited from B, and the complementation of Be has the form x 7→ e − x := e ∧ x∗ for all x ∈ B. An ideal J is proper provided that J 6= B. A regular ideal of B is often called a band or component of B.

Boolean Algebras and Vector Lattices

7

(3) Take Boolean algebras B and B 0 and a mapping h : B → B 0 . Say that h isotonic or monotone if (x ≤ y ⇒ h(x) ≤ h(y)). (Note by the way an isotonic mapping from B to B 0 with the opposite order is antitonic.) Say that h is a (Boolean) homomorphism, if for all x, y ∈ B the following equalities are fulfilled: h(x ∨ y) = h(x) ∨ h(y), h(x ∧ y) = h(x) ∧ h(y), h(x∗ ) = h(x)∗ . Every homomorphism h is monotone and the image h(B) of B is a subalgebra of B 0 . If h is bijective then we call h an isomorphism, and B and B 0 , isomorphic Boolean algebras. An injective homomorphism is a monomorphism. A homomorphism h (of B to a complete B 0 ) is complete if h preserves suprema and infima; i.e., h(sup(U )) = sup(h(U )) and h(inf(V )) = inf(h(V )) for all U ⊂ B and V ⊂ B for which there are sup(U ) and inf(V ). Observe that h : B → B 0 is a ring homomorphism (see 1.1.4) if and only if h is a Boolean homomorphism from B into the Boolean algebra Ba0 := [0, a] with a := h(1B ). Given a set C and a bijection h : B → C, we may furnish C with an order by putting h(x) ≤ h(y) whenever x ≤ y. In this event, C turns into a Boolean algebra and h becomes an isomorphism between B and C. (4) Let J be a proper ideal of a Boolean algebra B. Define the equivalence ∼ on B by the rule x∼y ⇔ xMy∈J

(x, y ∈ B).

Denote by ϕ the factor mapping of B onto the factor set B/J := B/∼. Recall that ϕ is also called canonical. Given cosets (equivalence classes) u and v, i.e., members of B/J; agree to write u ≤ v if and only if there are x ∈ u and y ∈ v satisfying x ≤ y. We have thus defined an order on B/J. In this event, B/J becomes a Boolean algebra which is called factor algebra or quotient algebra of B by J. The Boolean operations in B/J make ϕ a homomorphism. So, ϕ is referred to as the factor homomorphism of B onto B/J. If h : B → B 0 is a homomorphism then ker(h) := {x ∈ B : h(x) = 0} is an ideal of B and there is a unique monomorphism g : B/ ker(h) → B 0 satisfying g ◦ ϕ = h, where ϕ : B → B/ ker(h) is the factor homomorphism. Therefore, each homomorphic image of a Boolean algebra B is isomorphic to the factor algebra of B by a suitable ideal. B :=

Q

(5) Take a family of Boolean algebras (Bα )α∈A . Furnish the product α∈A Bα with the coordinatewise order or product order by putting x ≤ y for

8

Chapter 1

x, y ∈ B whenever x(α) ≤ y(α) for all α ∈ A. In this event B becomes a Boolean algebra. Each Boolean operation in B consists in implementing the respective operation in every coordinate Boolean algebra Bα , i.e., it is carried out coordinatewise. The zero 0B and unity 1B of B are as follows: 0B (α) := 0α and 1B (α) := 1α (α ∈ A), where 0α and 1α are the zero and unity in Bα . The Boolean algebra B is the Cartesian product or, simply, product of (Bα )α∈A . (6) A disjoint subset of a Boolean algebra is an antichain. In other words, a subset A of B is an antichain provided that a1 ∧ a2 = 0 for all distinct a1 , a2 ∈ A. If an antichain has the form A := {aξ : ξ ∈ „} then we presume that aξ ∧ aη = 0 whenever ξ 6= η. An antichain A in B is a partition of an element b ∈ B and a partition of W unity, when b is the unity of B, provided that b = A. Let (bξ )ξ∈„ be a partition of unity in B. According to 1.1.5 (2) Bξ := [0, bξ ] is a Boolean algebra with unity bξ . Q The complete Boolean algebra B is isomorphic to ξ∈„ Bξ . Some isomorphism is carried out by sending b ∈ B to the mapping eb by the rule eb(ξ) := b ∧ bξ (b ∈ B). 1.1.6. Recall a certain fundamental property of Boolean algebras. Let B be a Boolean algebra. A subset E of B minorizes a subset B0 of B if to each 0 < b ∈ B0 there is an x in E such that 0 < x ≤ b. It is also in common parlance to call E coinitial or minorant set to B0 . Denote by u.b.(M ) the set of all upper bounds of M . Exhaustion Principle. Let M be a nonempty subset of a Boolean algebra B. Assume given a subset E of B that minorizes the band B0 of B generated by M . Then some antichain E0 exists, E0 ⊂ E, such that u.b.(E0 ) = u.b.(M ) and to each x ∈ E0 there is an element y in M satisfying x ≤ y. C Consider the set A of all antichains A with the following properties: (a) A ⊂ E; (b) to each x ∈ A there is a y ∈ M satisfying x ≤ y. If 0 6= y ∈ M then, by the minorant condition, y ≥ x for some 0 6= x ∈ E. Hence, {x} ∈ A and A is nonempty. The inclusion ordered set A clearly obeys the hypotheses of the Kuratowski–Zorn Lemma. Therefore, there is a maximal element E0 ∈ A. The property (b) from the definition of A implies u.b.(M ) ⊂ u.b.(E0 ). In particular, we are done if u.b.(E0 ) = {1}. To prove the converse inclusion assume that b0 ∈ / u.b.(M ) for some b0 ∈ u.b.(E0 ), b0 6= 1. There exists an element x ∈ M , such that x0 := b∗0 ∧ x 6= 0. By the minorant condition, 0 < y ≤ x0 for some y ∈ E. The set E0 ∪ {y} belongs to A and has essentially more elements than E0 . This contradicts the fact that E0 is maximal, and so u.b.(E0 ) ⊂ u.b.(M ). B

Boolean Algebras and Vector Lattices

9

(1) To each nonempty set M ⊂ B havingWthe least W upper bound, there is an antichain A ⊂ B with the following properties: A = M and, given x ∈ A, we may find y in M such that x ≤ y. S C Choose the minorant E := y∈M [0, y] as a minorant for M and appeal to (1). B (2) A Boolean algebra is complete if and only if any antichain in it has the supremum. 1.1.7. Examples. (1) Recall that P(X) stands for the powerset of X comprising all subsets of X and also denoted by 2X . Given a nonempty set X, observe that the powerset P(X) of X ordered by inclusion is a complete Boolean algebra. This algebra is often called the boolean of X. The Boolean operations on every boolean are the set-theoretic operations of union, intersection, and complementation. (2) Let X be a topological space. Recall that a closed and open subset of X is called clopen. The collection of all clopen sets in X, ordered by inclusion, is a subalgebra of the boolean P(X). Denote this subalgebra by Clop(X). The Boolean operations in Clop(X) are inherited from P(X). Hence, they are settheoretic. However, Clop(X) is not a regular subalgebra of P(X); i.e., the infinite operations in P(X) and Clop(X) may differ essentially. (3) A closed subset F of a topological space X is called regular if F = cl(int(F )); i.e., if F coincides with the closure of the interior of F . By analogy, a regular open set G is defined by the formula G = int(cl(G). Let RC (X) and RO (X) stand for the collections of all regular closed subsets and all regular open subsets, respectively, of X. Furnished with the order by inclusion, RC (X) and RO (X) become complete Boolean algebras. The mapping F 7→ int(F ) (F ∈ RC (X)) is an isomorphism between them. Despite RC (X) and RO (X) are included in the boolean P(X), they are not subalgebras of the latter. For instance, the Boolean operations on RC (X) take the form E ∨ F = E ∪ F,

E ∧ F = cl(int(E ∩ F )),

F ∗ = cl(X − F ).

(4) Denote by Bor(X) the Borel σ-algebra of a topological space X (i.e., the σ-regular subalgebra of the boolean P(X) which is generated by the open sets of X). Consider the ideal N of Bor(X) comprising the meager subsets of X (also called the first category sets in X). The factor algebra Bor(X)/N is a complete Boolean algebra called the algebra of Borel sets modulo meager sets or briefly the Borel-by-meager algebra. We will arrive at an isomorphic algebra if instead of Bor(X) we take the σalgebra of sets with the Baire property. (A subset M of X has the Baire property if

10

Chapter 1

there is an open set G in X such that the symmetric difference M M G is a meager set.) If X is a Baire space, i.e., if X lacks nonempty open meager subsets, then the algebra in question is isomorphic to the algebra of regular closed sets RC (X). (5) Assume given a nonempty set Š, a σ-algebra B ⊂ P(Š) of subsets of Š, and a measure on B which is a positive countably additive function µ : B → R ∪ {+∞}, with R the reals. Countable additivity, as usual, means that [  X ∞ ∞ An = µ µ(An ) n=1

n=1

for every disjoint sequence (An ) of B. A triple (Š, B, µ) is said to be a measure space if the following conditions are met: (a) if A ⊂ Š and A ∩ K ∈ B for each K ∈ B with µ(K) < +∞, then A ∈ B; (b) if A ∈ B and µ(A) = +∞ then there is A0 ∈ B such that A0 ⊂ A and 0 < |A0 | < +∞; (c) if A ∈ B, µ(A) = 0 and A0 ⊂ A then A0 ∈ B. Let N := {A ∈ B : µ(A) = 0}. Then N is a σ-complete ideal. The factor algebra B(Š) := B(Š, B, µ) := B/N is also a σ-algebra called the associated algebra or the algebra of measurable sets by measure zero sets. The measure space (Š, B, µ) is said to possess the direct sum property if B contains a family (Aξ )ξ∈„ of pairwise disjoint sets of finite measure such that the following holds: for every measurable subset A ∈ B of finite measure there exists a countable set of indices ‚ ⊂ „ and a measure zero set A0 ∈ N such that [  A = A0 ∪ (A ∩ Aξ ) . ξ∈‚

If a measure space (Š, B, µ) possesses the direct sum property then the associate Boolean algebra B(Š, B, µ) is complete. 1.2. Representation of Boolean Algebras The Stone Representation Theorem opens up a distinct possibility of representing a Boolean algebra as the Boolean algebra of clopen subsets of a compact space. The basic goal of this section is to prove this theorem and to describe some opportunities it affords. 1.2.1. Let B be an arbitrary Boolean algebra. (1) A character of B is a Boolean homomorphism or, which is the same, a nonzero ring homomorphism χ : B → 2. Denote by X(B) the set of all

Boolean Algebras and Vector Lattices

11

characters of B and make X(B) into a topological space on furnishing it with the topology of pointwise convergence. To put it more explicitly, the topology on X(B) is induced by the product topology of 2B , where we consider 2 with the unique compact topology on this set, the discrete topology of 2. Observe that, unless stated otherwise, we presume all topological spaces to be Hausdorff. Recall that a topological space X is connected whenever the only clopen subsets of X are ∅ and X. A topological space X is totally disconnected provided that each connected subspace of X is at most a singleton. The topological space 2B , called sometimes a Cantor discontinuum, is compact and totally disconnected. A topological space with all these properties is a Boolean space. Evidently, X(B) is a closed subset of 2B . Therefore, X(B) is itself is a Boolean space. Say that the Boolean space X(B) is the character space of a Boolean algebra B. (2) Recall that a nonempty subset F of B is a filter on B provided that x ∈ F, y ∈ F ⇒ x ∨ y ∈ F, x ∈ F, x ≤ y ⇒ y ∈ F. A filter other than B is proper. A maximal element of the inclusion-ordered set of all proper filters on B is an ultrafilter on B. Let U (B) stand for the set of all ultrafilters on B, and denote by U (b) the set of ultrafilters containing b. We endow U (B) with the topology with base {U (b) : b ∈ B}. This definition is sound since it is easy to check that U (x ∧ y) = U (x) ∩ U (y) (x, y ∈ B); i.e., U (B) is closed under finite intersections. The topological space U (B) is often referred to as the Stone space of B and is denoted by S (B). (3) Denote by M (B) the set of all maximal (proper) ideals of a Boolean algebra B. An ideal here may be understood in accord with 1.1.5 (2) or in the conventional sense of ring theory. Clearly, a set J in B is an ideal of B if and only if J ∗ := {x∗ : x ∈ J} is a filter on B. Moreover, J ∈ M (B) ↔ J ∗ ∈ U (B). Therefore, the mapping J 7→ J ∗ presents a bijection between M (B) and U (B). The set M (B) is usually called the maximal ideal space of B and is always furnished with the inverse image topology translated from U (B) which makes the mapping J 7→ J ∗ a homeomorphism. 1.2.2. Recall the prerequisites we need for applying the Gelfand transform in the case of a Boolean algebra. (1) A Boolean ring B is a field if and only if B is the pair of 0 and 1. Hence, there is a unique Boolean field to within isomorphism; namely, 2. C Indeed, a nonzero element x ∈ B is invertible, and so the following implications are valid: xx−1 = 1 ⇒ xxx−1 = 1 ⇒ xx−1 = x ⇒ x = 1. B

12

Chapter 1

Given χ ∈ X(B), denote by χ∗ the mapping x 7→ χ(x)∗ (x ∈ B). Obviously, ker(χ) := {x ∈ B : χ(x) = 0} is an ideal, and ker(χ)∗ is a filter. (2) The mappings χ 7→ ker(χ) (χ ∈ X(B)) and χ 7→ ker(χ)∗ (χ ∈ X(B)) are homeomorphisms of X(B) onto M (B) and U (B), respectively. C The mapping χ 7→ ker(χ) is injective. If J ∈ M (B) then B/J is a field and, by (1), B/J is isomorphic to 2. Fix such an isomorphism λ : B/J → 2; and put χ := λ ◦ ϕ, where ϕ : B → B/J is the factor mapping. Obviously, ker(χ) = J and so the mapping under discussion is bijective. The remaining claims are obvious. B (3) For an x in B to equal zero it is necessary and sufficient that χ(b) = 0 for all χ ∈ X(B). C Assume that x 6= 0. Then the principal ideal {y ∈ B : y ≤ x∗ } is proper, and so can be extended to a maximal ideal J ∈ M (B). This claim, known as the Krull Theorem, is immediate from the Kuratowski–Zorn Lemma. By (2), J = ker(χ) for a certain χ ∈ X(B). Since x ∈ / J; therefore, χ(x) 6= 0. B 1.2.3. Stone Representation Theorem. Each Boolean algebra B is isomorphic to the Boolean algebra of clopen sets of a Boolean space unique up to homeomorphism, the Stone space of B. C Denote by C(X(B), 2) the algebra of continuous 2-valued functions on the character space X(B) of B which is a Boolean space. The Gelfand transform GB assigns to an element x ∈ B the 2-valued function x b : χ 7→ χ(x) (χ ∈ X(B)). Obviously, GB : B → C(X(B), 2) is a injective homomorphism, i.e., a monomorphism (cf. 1.2.2 (3)). Take f ∈ C(X(B), 2) and put Vf := {χ ∈ X(B) : f (χ) = 1}. The set Vf is clopen. By the definition of the topology of X(B), there are b1 , . . . , bk ∈ B and c1 , . . . , cl ∈ B such that Vf := {χ ∈ X(B) : χ(bn ) = 1 (n ≤ k),

χ(cm ) = 0 (m ≤ l)}.

Assign b0 := b1 ∧ · · · ∧ bk , c0 := c1 ∨ · · · ∨ cl and b := b0 ∧ c∗0 . The set Vf can be presented as follows: Vf = {χ ∈ X(B) : χ(b0 ) = 1, χ(c0 ) = 0} = {χ ∈ X(B) : χ(b) = 1} = {χ ∈ X(B) : bb(χ) = 1}. Therefore, f = bb, and so GB is an isomorphism. Assume now that Q1 and Q2 are Boolean spaces such that the mapping h : C(Q1 , 2) → C(Q2 , 2) is an isomorphism of these algebras. If χ is a character of

Boolean Algebras and Vector Lattices

13

C(Q2 , 2) then χ◦h is a character of C(Q2 , 2). Hence, χ 7→ χ◦h is a homeomorphism between the character spaces. On the other hand, the character space of C(Qk , 2) is homeomorphic to Qk . Therefore, the Boolean spaces Q1 and Q2 are homeomorphic. It suffices to note that the algebra C(X(B), 2) is isomorphic to the algebra of clopen sets of the space X(B) and so, of the space U (B) as well. B In view of this theorem, there is a mapping B 7→ Clop(S (B)) which is occasionally called the Stone transform of B. 1.2.4. In the sequel we are mostly interested in complete and σ-complete Boolean algebras. The notion of a complete Boolean algebra is closely tied with that of an extremally disconnected compact space. Recall that a topological space Q is called extremally (quasiextremally) disconnected or simply extremal (quasiextremal ) if the closure of an arbitrary open set (open Fσ -set) in it is open or, which is equivalent, the interior of an arbitrary closed set (closed Gδ -set) is closed. Clearly, an extremal (quasiextremal) space is totally disconnected. Ogasawara Theorem. A Boolean algebra is complete (σ-complete) if and only if its Stone space is extremal (quasiextremal). C We confine exposition to the case of complete Boolean algebra B. Assume further that h is an isomorphism of B on the algebra of clopen sets of the compact spaceSQ := U (B). Take an open set G ⊂ Q. Since Q is totally disconnected, G = U , where U stands for W the union of all clopen subsets of G. Put U 0 := −1 {h (U ) : U ∈ U } and b := U 0 . The clopen set h(b) is the closure of G. Indeed, cl(G) ⊂ h(b) and h(b)\ cl(G) is open. If the latter set is nonempty then h(c) ⊂ h(b)\ cl(G) for some 0 6= c ∈ B. This impliesWin turn that h(c) ∨ h(u) ≤ h(b) for all u ∈ U 0 , which contradicts the equality b = U . Therefore, cl(G) = h(b) is an open set. Assume now that the compact space Q is extremal. Let G stand for some S collection of clopen subsets of Q, and put G := G . The set G is open and the closure cl(G) of G must be open by the hypothesis about Q. Obviously, cl(G) is the least upper bound of G in the Boolean algebra of clopen sets Clop(Q). B 1.2.5. Sikorski Theorem. Assume that B and B 0 are Boolean algebras, and h : B → B 0 is a homomorphism between them. Denote by ı : B → Clop(S (B)) and ı0 : B 0 → Clop(S (B 0 )) the Stone transforms of B and B 0 . There is a unique continuous mapping θ : S (B 0 ) → S (B) such that h(x) = (ı0 )−1 θ−1 (ı(x)) (x ∈ B). The mapping h 7→ S (h) := θ carries out a bijection between the sets of all homomorphisms from B to B 0 and the set of all continuous mappings from S (B 0 ) to S (B). If B 00 is another Boolean algebra and g : B 0 → B 00 is a homomorphism, then S (g ◦ h) = S (h) ◦ S (g). Moreover, S (IB ) = IS (B) .

14

Chapter 1

C Denote Q := S (B) and Q0 := S (B 0 ). If q 0 is an ultrafilter in B 0 then obviously q := {b ∈ B : h(b) ∈ q 0 } is also an ultrafilter in B. Assigning θ(q 0 ) := q we arrive at the mapping θ : q 0 ∈ Q0 7→ q ∈ Q. Taking into consideration the equalities ı(b) = {q ∈ Q : b ∈ q} (b ∈ B) and ı0 (b0 ) = {q 0 ∈ Q0 : b0 ∈ q 0 } (b0 ∈ B 0 ) we deduce ı0 h(b) = {q 0 ∈ Q0 : h(b) ∈ q 0 } = {q 0 ∈ Q0 : b ∈ q} = {q 0 ∈ Q0 : θ(q 0 ) ∈ ı(b)} = θ−1 (ı(b)). In particular, θ(q 0 ) ∈ ı(b) if and only if q 0 ∈ ı0 (h(b)); therefore θ is continuous. The remaining properties of θ are obvious. B Two important particular cases of the situation under consideration are worthy of a special attention. (1) A Boolean algebra B0 is isomorphic to a subalgebra of a Boolean algebra B if and only if the Stone space S (B0 ) of B0 is a continuous image of the Stone space S (B) of B. (2) A Boolean algebra B 0 is a homomorphic image of a Boolean algebra B (or isomorphic with a factor algebra of B; see 1.1.6 (4)) if and only if the Stone space S (B 0 ) of B 0 is homeomorphic to a closed subset of the Stone space S (B) of B. 1.2.6. Loomis–Sikorski Theorem. Let Q be the Stone space of a Boolean σ-algebra B. Denote by Clopσ (Q) the σ-algebra of subsets of Q which is generated by the set Clop(Q) of all clopen subsets of Q. Let  stand for the σ-ideal of Clopσ (Q), comprising all meager sets. Then B is isomorphic with the factor algebra Clopσ (Q)/. If ı is an isomorphism of B onto Clop(Q) and ϕ is the factor mapping of Clopσ (Q) onto the factor algebra Clopσ (Q)/ then the mapping h := ϕ ◦ ı is an isomorphism of B onto Clopσ (Q)/. C Observe that h is a homomorphism as the composite of two homomorphisms. If h(b) = 0 then ı(b) ∈  and ı(b) = ∅, since no nonempty clopen set is meager. Thus h is injective. To prove that h is surjective put F := {A ∈ Clopσ (Q) : (∃b ∈ B) ϕ(A) = h(A)}. Since Clop(Q) ⊂ F ⊂ Clopσ (Q), it suffices to observe that F is a σ-algebra. If A ∈ F then ϕ(Q \ A) = h(b∗ ), so that Q \ A ∈ F . Now, consider a sequence (An ) of F and choose a sequence W∞ S∞(bn ) of B such that ϕ(An ) = h(bn ). According to 1.1.7 (2) ı ( n=1 bn ) = A0 ∪ n=1 ı(bn ) with a nowhere-dense subset A0 ⊂ Q.

Boolean Algebras and Vector Lattices

15

Using this equality we easily deduce ! ! ! ∞ ∞ ∞ [ [ [ ϕ An = ϕ A0 ∪ An = ϕ A0 ∪ ı(bn ) n=1

=ϕ ı

n=1 !! ∞ _

bn

n=1

n=1

=h

∞ _

! bn

n=1

and the result follows. B Birkhoff–Ulam Theorem. Let Q be a compact space. For every Borel set V ∈ Bor(Q) there exists a unique regular open set h(V ) such that the symmetric difference V 4h(V ) is meager. Let N stand for the σ-ideal of Bor(Q), comprising all meager sets. Then h is an order σ-continuous homomorphism from Bor(Q) onto RO (Q). Moreover, the kernel of h coincides with N and Bor(Q)/N is isomorphic to RO (Q). 1.2.7. Consider a measure space (Š, B, µ), and let ϕ : B → B(Š) be the factor homomorphism. A Boolean homomorphism ρ : B(Š) → B is called a lifting of the factor algebra B(Š) if ρ(A) ∈ A for each equivalence class A ∈ B(Š). The latter means that ϕ ◦ ρ is the identity mapping on B(Š); therefore, a lifting is a right-inverse of the homomorphism ϕ. (1) If a measure space (Š, B, µ) possesses the direct sum property then the factor algebra B(Š, B, µ) admits a lifting. C The proof can be found in [375]. B (2) Let ρ be a lifting of the factor algebra S B(Š). Then, for every family (Aξ )ξ∈„ T of elements of B(Š), the union As := ξ∈„ ρ(Aξ ) and the intersection Ai := ξ∈„ ρ(Aξ ) are measurable and, moreover, _ ^ ϕ(As ) = Aξ , ϕ(Ai ) = Aξ . ξ∈„

ξ∈„

(3) For every point ω ∈ Š, denote the ultrafilter {A ∈ B(Š) : ω ∈ ρ(A)} by τ (ω). The mapping τ : Š → Q thus constructed will be called the canonical immersion of Š in Q corresponding to the lifting ρ. 1.2.8. Theorem. Let ρ be a lifting of B(Š, B, µ), let τ be the corresponding canonical immersion of Š in the Stone space Q of the Boolean algebra B(Š, B, µ), and let ı be the Stone transform from B(Š) onto Clop(Q). The following are valid: (1) ρ(A) = τ −1 (ı(A)) for each class A ∈ B(Š); (2) ı−1 (U ) = ϕ(τ −1 (A)) for each clopen set U ∈ Clop(Q); (3) the mapping τ : Š → Q is Borel measurable and the image τ (Š) is dense in Q;

16

Chapter 1

(4) the inverse image τ −1 (N ) of every meager subset N ⊂ Q is measurable in Š and has zero measure. C Assertions (1) and (2) are straightforward. To prove (3) and (4), we consider an arbitrary open subset V ⊂ Q. Choose a family (Uξ )ξ∈„ of clopen subsets with S V = ξ∈„ Uξ . Then (1) implies measurability of the inverse images τ −1 (Uξ ) for all S −1 −1 ξ ∈ „ and 1.2.7 (2) implies measurability of the set τ (V ) = (Uξ ). The ξ∈„ τ W relation ξ∈„ Uξ = cl(V ) in the Boolean algebra Clop(Q), together with (2) and 1.2.7 (2), ensures the equality ϕ(τ −1 (V )) = ϕ(τ −1 (cl V ). B 1.2.9. A function µ : B → R ∪ {+∞} is called additive, countably additive, or completely additive if _  X µ xξ = µ(xξ ) ξ∈„

ξ∈„

for any finite, countable, or arbitrary antichain (xξ ) of B, respectively. To eliminate the trivial additive function µ ≡ +∞ it is always assumed that µ(0B ) = 0. An additive function on Boolean algebra is often referred to as a measure. We say that the function µ is positive, strictly positive, and finite if respectively µ(b) ≥ 0, µ(b) > 0, and µ(b) < +∞ for all 0 6= b ∈ B. Finally, a positive µ is said to be locally finite if for any b ∈ B with 0 < µ(b) there exists 0 < b0 ≤ b such that 0 < µ(b0 ) < +∞. A positive countably additive locally finite measure on the Boolean algebra Clopσ (Q) is called normal if it vanishes on the ideal of meager sets. An extremal space is said to be hyperstonian if there is a normal measure on Clopσ (Q) strictly positive on Clop(Q). Let M+ (B) denote the set of all finite completely additive positive measures on B. A complete Boolean algebra B is said to be multinormed if the set of all finite completely additive measures separates the points of B; in symbols, (∀0 6= b ∈ B) (∃µ ∈ M+ (B)) µ(b) > 0. In the case when there is a finite strictly positive completely additive measure µ on B, the pair (B, µ) is referred to as a normed Boolean algebra. A normed Boolean algebra (B, µ) can be endowed with the metric ρ(x, y) := µ(x M y) and it is not difficult to observe that the metric space (B, ρ) is complete. 1.2.10. Theorem. For a complete Boolean algebra B the following are equivalent: (1) B is a multinormed Boolean algebra; (2) B is isomorphic to the Cartesian product of a family of normed Boolean algebras; (3) there exists a strictly positive locally finite completely additive measure on B;

Boolean Algebras and Vector Lattices

17

(4) B is isomorphic to the associated algebra B(Š, B, µ) for some measure space (Š, B, µ) possessing the direct sum property; (5) the Stone space S (B) is hyperstonian. C (1) ⇒ (2): According to the Exhaustion Principle (see 1.1.6) we may choose a partition of unity (bξ )ξ∈„ in B and a family of positive completely additive measures (µξ )ξ∈„ such that µξ (bξ ) > 0 for all ξ ∈ „. If Bξ is the principal ideal generated by bξ and the restriction of µξ to Bξ is denoted by the same symbol then (Bξ , µξ ) is a normed Boolean algebra and B is isomorphic to the Cartesian product of the
family (Bξ , µξ ) ξ∈„ . (2) ⇒ (3): The measure on B with the required properties can be defined by µ(b) :=

X

µξ (b ∧ bξ )

(b ∈ B).

ξ∈„

(3) ⇒ (4): Observe first that if Š := S (B) then Clopσ (Š) consists of the sets U 4N where U ∈ Clop(Š) and N ⊂ Š is meager. Now, let ν be a strictly positive locally finite completely additive measure on Clop(Š). If B := Clopσ (Š) and the measure µ on B is defined by letting µ(U 4N ) := ν(U ) then (Š, B, µ) is a measure space possessing the direct sum property and Boolean algebras B(Š, B, µ) and Clop(Š) are isomorphic, see [162]. (4) ⇒ (5): According to 1.2.8 a normal measure µ ¯ on Clopσ (Q) which is strictly ¯(A) := µ(τ −1 (A)) (a ∈ Clopσ (Q)). positive on Clop(Q) can be obtain by putting µ (5) ⇒ (1): Assume that Q := S (B) is hyperstonian, and let µ be a normal measure on Clopσ (Q) strictly positive on Clop(Q). Take an arbitrary b ∈ B. Since µ is locally finite there is a clopen set V with µ(V ) > 0 and V ⊂ ı(b) where ı : B → Clop(Q) is the Stone transform. Putting µb (x) := µ(V ∩ ı(x)) (x ∈ B) we arrive at a finite positive completely additive measure µb on B with µb (b) = µ(V ) > 0. B 1.3. Vector Lattices In this section we give some preliminaries to the theory of vector lattices; a more explicit exposition may be found elsewhere [15, 23, 145, 162, 163, 262, 336, 341, 388, 409]. 1.3.1. Let F be a linearly ordered field. An ordered vector space over F is a pair (E, ≤), where E is a vector space over F and ≤ is an order in E satisfying the following conditions: (1) if x ≤ y and u ≤ v then x + u ≤ y + v whatever x, y, u, v ∈ E might be;

18

Chapter 1

(2) if x ≤ y then λx ≤ λy for all x, y ∈ E and 0 ≤ λ ∈ F. Informally speaking, we may “sum inequalities in E and multiply them by positive members of F.” This circumstance is worded as follows: ≤ is an order compatible with vector space structure or, briefly, ≤ is a vector order. Equipping a vector space E over F with some vector order is equivalent to indicating a set E+ ⊂ E, called the positive cone of E, with the following properties: E+ + E+ ⊂ E+ ,

λE+ ⊂ E+ (0 ≤ λ ∈ F),

E+ ∩ −E+ = {0}.

Moreover, the order ≤ and the cone E+ are connected by the relation x ≤ y ⇔ y − x ∈ E+

(x, y ∈ E).

The elements of E+ are called positive. An ordered vector space E is called Archimedean if for any pair of elements x, y ∈ E the relation (∀n ∈ N) nx ≤ y implies x ≤ 0. In the sequel, all ordered vector spaces are assumed Archimedean. 1.3.2. A vector lattice is an ordered vector space that is also a lattice. Thereby in a vector lattice there exist a least upper bound sup{x1 , . . . , xn } := x1 ∨ · · · ∨ xn and a greatest lower bound inf{x1 , . . . , xn } := x1 ∧ · · · ∧ xn for every finite set {x1 , . . . , xn } ⊂ E. In particular, every element x of a vector lattice has the positive part x+ := x ∨ 0, the negative part x− := (−x)+ := −x ∧ 0, and the modulus |x| := x ∨ (−x). Let E be a vector lattice. For all x, y, z ∈ E the following relations are valid: (1) x = x+ − x− , |x| = x+ + x− = x+ ∨ x− ; (2) x ≤ y ⇔ x+ ≤ y + & y − ≤ x− ; (3) x ∨ y = 21 (x + y + |x − y|), x ∧ y = 12 (x + y − |x − y|); (4) |x| ∨ |y| = 12 (|x + y| + |x − y|), |x| ∧ |y| = 21 (|x + y| − |x − y|); (5) x + y = x ∨ y + x ∧ y, |x − y| = x ∨ y − x ∧ y; (6) x + y ∨ z = (x + y) ∨ (x + z), x + y ∧ z = (x + y) ∧ (x + z); (7) x, y, z ∈ E+ ⇒ (x + y) ∧ z ≤ (x ∧ z) + (y ∧ z); (8) |x − y| = |x ∨ z − y ∨ z| + |x ∧ z − x ∧ z|. Let (xα ) and (yα ) be families in E for which sup(aα ) and inf(yα ) exist. Then for any z ∈ E the infinite distributive laws are valid: (9) z ∧ supα (xα ) = supα (z ∧ xα ), z ∨ inf α (yα ) = inf α (z ∨ yα ). For the same (xα ), (yα ), and z the following useful relations are also true;

Boolean Algebras and Vector Lattices

19

(10) z + supα (xα ) = supα (z + xα ); (11) z + inf α (yα ) = inf α (z + yα ); (12) supα (xα ) = − inf α (−xα ). An order interval in E is a set of the form [a, b] := {x ∈ E : a ≤ x ≤ b}, where a, b ∈ E. The following constantly used property of vector lattices is frequently called the Riesz Decomposition Property. (13) [0, x + y] = [0, x] + [0, y] (x, y ∈ E+ ); We indicate only two corollaries of (13). (14) (x1 + · · · + xn ) ∧ y ≤ x1 ∧ y + · · · + xn ∧ y (xk , y ∈ E+ ); Vn Pm P (15) k=1 l=1 xk,l ≤ j∈J x1,j(1) ∧ · · · ∧ xn,j(n) where xk,l ∈ E+ and J is the set of all functions j : {1, . . . , n} → {1, . . . , m}. 1.3.3. Two elements x and y are called disjoint if |x|∧|y| = 0. The disjointness of x and y is denoted by x ⊥ y. (1) The following properties of disjointness are easy from 1.3.2: x ⊥ y ⇔ |x + y| = |x − y| ⇔ |x| ∨ |y| = |x| + |y|; x+ ⊥ x− ; (x − x ∧ y) ⊥ (y − x ∧ y); x ⊥ y ⇒ |x + y| = |x| + |y|, (x + y)+ = x+ + y + , (x + y)− = x− + y − . Let u ∈ E+ and e ∧ (u − e) = 0 for some 0 ≤ e ∈ E. Then e is said to be a fragment, or a part, or a component of u, or a unit element with respect to u. (2) The set E(u) of all fragments of u with the order induced by E is a Boolean algebra. The lattice operations in E(u) are taken from E and the Boolean complement has the form e∗ := u − e (e ∈ E(u)). The disjoint complement M ⊥ of a nonempty set M ⊂ E is defined as M ⊥ := {x ∈ E : (∀y ∈ M ) x ⊥ y}. A nonempty set K in E meeting the identity K = K ⊥⊥ is called a band (a component in the Russian literature) of E. Every band of the form {x}⊥⊥ with x ∈ E is called principal. (3) The inclusion-ordered set of all bands of E is denoted by B(E) and presents a complete Boolean algebra. The Boolean operations of B(E) take the shape: L ∧ K = L ∩ K,

L ∨ K = (L ∪ K)⊥⊥ ,

L∗ = L⊥

(L, K ∈ B(E)).

20

Chapter 1

The Boolean algebra B(E) is the base of E. Let K be a band of the vector lattice E. If there is an element sup{u ∈ K : 0 ≤ u ≤ x} in E then it is called the band projection of x onto K and is denoted by [K]x (or πK x). Given an arbitrary x ∈ E, we put [K]x := [K]x+ − [K]x− . The band projection of an element x ∈ E onto K exists if and only if x is representable as x = y + z with y ∈ K and z ∈ K ⊥ . Furthermore, y = [K]x and z = [K ⊥ ]x. Assume that to each element x ∈ E there is a band projection onto K, then the operator x 7→ [K]x (x ∈ E) is a linear idempotent and 0 ≤ [K]x ≤ x for all 0 ≤ x ∈ E, called a band projection or an order projection. (4) The set P(E) of all band projections ordered by π ≤ ρ ⇔ π ◦ ρ = π is a Boolean algebra. The Boolean operations of P(E) take the shape π ∧ ρ = π ◦ ρ,

π ∨ ρ = π + ρ − π ◦ ρ,

π ∗ = IE − π

(π, ρ ∈ (E)).

The band projection onto a principal band is called principal. (5) The principal projection πu := [u] := [u⊥⊥ ], where 0 ≤ u ∈ E, can be calculated by the following rule simpler than that indicated above: πu x = sup{x ∧ (nu) : n ∈ N}. A vector lattice E is said to have the projection property (principal projection property) if every band (every principal band) in B(E) is a projection band. Clearly, every Kσ -space has the principal projection property. 1.3.4. The order relation in a vector lattice generates different types of convergence. Let (A, ≤) be an upward-directed set. A net (xα ) := (xα )α∈A in E is called increasing (decreasing) if xα ≤ xβ (xβ ≤ xα ) for α ≤ β (α, β ∈ A). We say that a net (xα ) in a vector lattice E o-converges to x ∈ E if there exists a decreasing net (eβ )β∈B in E such that inf{eβ : β ∈ B} = 0 and for each β ∈ B there is α(β) ∈ A with |xα − x| ≤ eβ (α ≥ α(β)). In this event, we call x the (o)

o-limit of the net (xα ) and write x = o-lim xα or xα → x. If a net (eβ ) in this definition is replaced by a sequence (λn e)n∈N , where 0 ≤ v ∈ E+ and (λn )n∈N is a numerical sequence with limn→∞ λn = 0, then we say that a net (xα )α∈A converges relatively uniformly or more precisely e-uniformly to x ∈ E. The elements e and x are called the regulator of convergence and the r-limit (r)

of (xα ), respectively. The notations x = r-limα∈A xα and xα → x are also frequent. A net (xα )α∈A is called o-fundamental (r-fundamental with regulator e) if the net (xα − xβ )(α,β)∈A×A o-converges (respectively, r-converges with regulator e) to zero. A vector lattice is said to be relatively uniformly complete if every rfundamental sequence is r-convergent.

Boolean Algebras and Vector Lattices

21

The presence of order convergence in a vector lattice allows us to determined the sum of an infinite family (xξ )ξ∈„ . Indeed, given θ := {ξ1 , . . . , ξn } ∈ Pfin („), put yθ := xξ1 + . . . + xξn . So, we arrive at the (yθ )θ∈‚ , where ‚ := Pfin („) is naturally ordered by inclusion. Assuming that there is some x satisfying x = o-limθ∈‚ yθ , we call the family (xξ ) summable in order or order summable or o-summable. The P element x is the o-sum of (xξ ) and we write x = o- ξ∈„ xξ . Obviously, if xξ ≥ 0 (ξ ∈ „) then for the o-sum of the family (xξ ) to exist it isPnecessary and sufficient that the net (yθ )θ∈‚ has the supremum, in which case o- ξ∈„ xξ = supθ∈‚ yθ . If (xξ ) is a disjoint family then X − oxξ = sup x+ ξ − sup xξ . ξ∈„

ξ∈„

ξ∈„

1.3.5. (1) A linear subspace J of a vector lattice is called an order ideal or o-ideal (or, finally, just an ideal, when it is clear from the context what is meant) if the inequality |x| ≤ |y| implies x ∈ J for arbitrary x ∈ E and y ∈ J. Every order ideal of a vector lattice is a vector lattice. If an ideal J possesses the additional property J ⊥⊥ = E (or, which is the same, J ⊥ = {0}) then J is referred to as an order-dense ideal of E (the term “foundation” is current in the Russian literature). (2) Let J be an ideal of a vector lattice E. Then the factor space e e is determined by the E := E/J is also a vector lattice, provided that the order on E e standing for the canonical factor mapping. positive cone ϕ(E+ ), with ϕ : E → E The factor lattice E/J is Archimedean if and only if N is closed under relative uniform convergence. If E is an f -algebra and J is a ring and order ideal then E/N is an f -algebra and ϕ is algebra homomorphism. If E is a Kσ -space and J is sequentially order-closed then E/J is a Kσ -space and ϕ is sequentially order continuous. (3) Denote by I (E) the set of all order ideals of E ordered by inclusion. Then I (E) is a complete lattice, with the lattice operations defined as I ∧J := I ∩J and I ∨J := I +J. Moreover, the lattice I (E) is distributive. The sublattice Ip (E) of principal ideals is also distributive. 1.3.6. (1) A vector sublattice is a vector subspace E0 ⊂ E such that x ∧ y, x ∨ y ∈ E0 for all x, y ∈ E0 . We say that a sublattice E0 is minorizing if, for every 0 6= x ∈ E+ , there exists an element x0 ∈ E0 satisfying the inequalities 0 < x0 ≤ x. We say that E0 is a majorizing or massive sublattice if, for every x ∈ E, there exists x0 ∈ E0 such that x ≤ x0 . Thus, E0 is a minorizing or majorizing sublattice if and only if E+ \ {0} = E+ + E0 + \ {0} and E = E+ + E0 , respectively. (2) A set in E is called (order ) bounded (or o-bounded ) if it is included in some S order interval. The o-ideal generated by the element 0 ≤ u ∈ E is the set ∞ E(u) := n=1 [−nu, nu]; clearly, E(u) is the smallest o-ideal in E containing u.

22

Chapter 1

If E(u) = E then we say that u is a strong unity or strong order-unity and E is a vector lattice of bounded elements. If E(u)⊥⊥ = E then we say that u is an order-unity or weak order-unity. It is evident that an element u ∈ E+ is an order-unity if {u}⊥⊥ = E; i.e., if E lacks nonzero elements disjoint from u. (3) An element x ≥ 0 of a vector lattice is called discrete if [0, x] = [0, 1]x; i.e., if 0 ≤ y ≤ x implies y = λx for some 0 ≤ λ ≤ 1. A vector lattice E is called discrete or atomic if, for every 0 6= y ∈ E+ , there exists a discrete element x ∈ E such that 0 < x ≤ y. If E lacks nonzero discrete elements then E is said to be continuous or diffuse. 1.3.7. A vector lattice is said to be (conditionally) order complete if each nonvoid order bounded set in it has least upper and greatest lower bounds. If, in a vector lattice, least upper and greatest lower bounds exist only for countable bounded sets, then it is called countably order complete. An order complete vector lattice and a countably order complete vector lattice are frequently referred to as a Dedekind complete vector lattice and a Dedekind σ-complete vector lattice or, in the Russian literature, K-space(= Kantorovich space) and a Kσ -space, respectively. We say that a K-space (Kσ -space) is universally complete or extended if its every subset (countable subset) of pairwise disjoint elements is bounded. (1) Theorem. Let E be an arbitrary K-space. Then E has the projection property and the operation of projecting onto bands determines the isomorphism K 7→ [K] of the Boolean algebras B(E) and P(E). If there is an order-unity 1 in E then the mappings π 7→ π1 from P(E) into E(E) and e 7→ {e}⊥⊥ from E(E) into B(E) are isomorphisms of Boolean algebras, too. A K-space is o-complete in the sense that every o-fundamental net in it is oconvergent. Each Kσ -space and, hence, a K-space is Archimedean. Henceforth all vector lattices are presumed to be Archimedean. Consider an order-bounded net (eα )α∈A in a K-space E , and let e ∈ E. (2) An order-boundednet (eα )α∈A o-converges to e if and only if the relation o-limα∈A [d] (|eα − e| − d)+ = 0 holds in the Boolean algebra P(E) for all positive d ∈ E. C It is easy to verify the necessity of the criterion in question. To prove its sufficiency, assign e0 := inf α∈A supβ>α |eβ − e|. If the net (eα )α∈A does not converge to e then e0 > 0 and, thus, there are π ∈ P(E), d ∈ D, and n ∈ N such that 0 < πd/n < e0 . Therefore, for each index α ∈ A, we have h i   sup [d] |eβ − e| > d/n = [d] sup |eβ − e| > d/n > π, β>α

β>α

  which contradicts the convergence of [d] |eα − e| > d/n to zero. B

Boolean Algebras and Vector Lattices

23

(3) Suppose that E is a K-space with order-unity 1; while (eα )α∈A is a bounded net in E, and e ∈ E. Then o-limα∈A eα = e if and only if the relation  o-limα∈A (|eα − e| − 1/n)+ = 0 holds in the Boolean algebra P(E) for all n ∈ N. 1.3.8. Let E be a Kσ -space with order-unity 1. We call the projection of the order-unity to the band {x}⊥⊥ the trace of x and denoted it by ex . Therefore, ex := sup{1 ∧ (n|x|) : n ∈ N}. The trace ex serves both as an order-unity of {x}⊥⊥ and a unit element of E. Given a real λ, denote the trace of the positive part of λ1 − x by exλ ; i.e., exλ := e(λ1−x)+ . The function λ 7→ exλ (λ ∈ R) arising in this case is called the spectral function or characteristic of x. Theorem. Let E be an arbitrary Kσ -space with order-unity 1 and P be a dense subfield of R. The spectral function λ 7→ exλ (λ ∈ R) of x ∈ E has the following properties: (1) (∀ λ, µ ∈ R) (λ ≤ µ ⇒ exλ ≤ exµ ); V W (2) ex+∞ := µ∈P exµ = 1, ex−∞ := µ∈P exµ = 0; W (3) exλ = {exµ : µ ∈ P, µ < λ} (λ ∈ R); (4) x ≤ y ⇔ (∀ λ ∈ P) (eyλ ≤ exλ ); W (5) ex+y = {exµ ∧ eyν : µ, ν ∈ P, µ + ν = λ}; λ W x (6) ex·y {eµ ∧ eyν : 0 ≤ µ, ν ∈ P, µν = λ} (x ≥ 0, y ≥ 0); λ = W (7) e−x {1 − ex−µ : µ ∈ P, µ < λ} = (1 − ex−λ ) · e(x+λ1) ; λ = (8) ex∧y = exλ ∨ eyλ ; ex∨y = exλ ∧ eyλ (λ ∈ R); λ λ W (9) x = inf(A) ⇔ (∀ λ ∈ P) (exλ = {eaλ : a ∈ A}); |x|

(10) eλ = exλ ∧ (1 − ex−λ ) ∧ ex+λ1

(λ ∈ R);

−x x αx (11) eαx λ = eλ/α (α > 0), eλ = e−λ/α (α < 0)

(λ ∈ R);

x ∗ cx x (12) ecx λ = c ∧ eλ + c (λ > 0), eλ = c ∧ eλ (λ ≤ 0)

(c ∈ E(E)).

A formula for x = sup(A) similar to (9) is generally not true. Nevertheless the following more complicated formula is valid:  x = sup(A) ⇔ (∀ λ ∈ P)

exλ

=

_^

{eaν

 : a ∈ A} .

ν<λ

If the function λ 7→ inf{eaλ : a ∈ A} satisfies (3) then the spectral function for x = sup(A) is calculated by (9) in which sup and inf are interchanged.

24

Chapter 1

1.3.9. (1) Freudenthal Spectral Theorem. Let E be an arbitrary Kσ -space with an order-unity 1. Every element x ∈ E admits the representation Z∞ x=

λ dexλ ,

−∞

where the integral is understood to be the 1-uniform limit of the integral sums X x(β) := τn (extn+1 − extn ), tn ≤ τn ≤ tn+1 , n∈Z

as δ(β) := supn∈Z (tn+1 − tn ) → 0, where β := (tn )n∈Z is a partition of the real line. In particular, the Freudenthal Spectral Theorem states that if E is a Kσ -space and e ∈ E+ , then every element x ∈ E(e) can be e-uniformly approximated by Pn linear combinations of fragments of e, i.e. by elements of the form k=1 λk ek , where λ1 , . . . , λn ∈ R and e1 , . . . , en ∈ E(e). In the case when the latter holds in a vector lattice E we say that E possesses the weak Freudenthal property. It may happen that every Pn element x ∈ E(e) can be e-uniformly approximated by linear combinations k=1 λk πk e, where λ1 , . . . , λn ∈ R and π1 , . . . , πn ∈ P(E). Then a vector lattice E is said to possesses the strong Freudenthal property. To characterize the spaces with strong Freudenthal property we need a definition. Two elements of a vector lattice E are completely disjoint if they lie in two disjoint projection bands of E. Clearly, a vector lattice with the principal projection property possesses the strong Freudenthal property. The converse is false. (2) Theorem. A vector lattice E possesses the strong Freudenthal property if and only if any two disjoint elements in E are completely disjoint. A vector lattice E possesses the weak Freudenthal property if and only if for any two elements e, d ∈ E there exist disjoint elements e0 ∈ [0, e] and d0 ∈ [0, d] with E(e + d) = E(e0 + d0 ). 1.3.10. An ordered algebra over F is an ordered vector space E over F which is simultaneously an algebra over the same field and satisfies the following condition: if x ≥ 0 and y ≥ 0 then xy ≥ 0 whatever x, y ∈ E might be. To characterize the positive cone E+ of an ordered algebra E, we must add to what was said in 1.3.1 the property E+ · E+ ⊂ E+ . We say that E is a lattice-ordered algebra if E is a vector lattice and an ordered algebra simultaneously. A lattice-ordered algebra is an f -algebra if, for all a, x, y ∈ E+ , the condition x ⊥ y = 0 implies that (ax) ⊥ y and (xa) ⊥ y. An f -algebra is called faithful or exact if, for arbitrary elements x and y, xy = 0 implies x ⊥ y. It is easy to show that an f -algebra is faithful if and only if it lacks nonzero nilpotent elements. The faithfulness of an f -algebra is equivalent to absence of any strictly positive element with zero square.

Boolean Algebras and Vector Lattices

25

1.3.11. A complex vector lattice is defined to be the complexification E ⊕ iE (with i standing for the imaginary unity) of a real vector lattice E. Often it is additionally required that the modulus |z| := sup{Re(eiθ z) : 0 ≤ θ ≤ π} exists for every element z ∈ E ⊕iE. In the case of a K-space or an arbitrary Banach lattice this requirement is automatically satisfied, since a complex K-space is the complexification of a real K-space. Speaking about order properties of a complex vector lattice E ⊕ iE, we mean its real part E. The concepts of sublattice, ideal, band, projection, etc. are naturally translated to the case of a complex vector lattice by appropriate complexification. 1.4. Representation of Vector Lattices In this section, we give some most important examples of function vector lattices. The question is studied of representing general vector lattices as those of measurable or continuous functions. 1.4.1. A special role in the theory of vector lattices is played by the spaces of continuous functions assuming possibly infinite values on a nowhere-dense set depending on a function. Before introducing these spaces, we need some preliminaries. Given a function f : Q → R and λ ∈ R, put {f < λ} := {t ∈ Q : f (t) < λ},

{f ≤ λ} := {t ∈ Q : f (t) ≤ λ}.

(1) Assume that Q is a topological space, ƒ is a dense set in R, and λ 7→ Uλ (λ ∈ ƒ) is an increasing mapping from ƒ to the inclusion ordered set P(Q). Then the following are equivalent: (a) There is a unique continuous function f : Q → R satisfying {f < λ} ⊂ Uλ ⊂ {f ≤ λ} (λ ∈ ƒ), (b) If λ, µ ∈ ƒ, and λ < µ then cl(Uλ ) ⊂ int(Uµ ). C The implication (a) ⇒ (b) is evident. Prove (b) ⇒ (a). To this end, given t ∈ Q, put f (t) := inf{λ ∈ ƒ : t ∈ Uλ }. For the so-defined f : Q → R, we easily see that {f < λ} ⊂ Uλ ⊂ {f ≤ λ}. It is also clear that [ \ {f < λ} = {Uµ : µ < λ & µ ∈ ƒ}, {f ≤ λ} = {Uν : λ < ν & ν ∈ ƒ}.

26

Chapter 1

Note that by now we have used only the fact that λ 7→ Uλ is an increasing mapping. Consider the mappings λ 7→ Vλ := int(Uλ ),

λ 7→ Wλ := cl(Uλ )

(λ ∈ ƒ).

These mappings are also increasing. So, the above implies that there are functions g and h : Q → R such that {g < λ} ⊂ Vλ ⊂ {g ≤ λ},

{h < λ} ⊂ Wλ ⊂ {h ≤ λ} (λ ∈ ƒ).

From the definition of Wλ it follows that Uµ ⊂ Wλ for µ < λ. Since ƒ is dense in R, to all t ∈ Q and ν > f (t) there are λ, µ ∈ ƒ such that f (t) < µ < λ < ν, so that t ∈ Uµ ⊂ Wλ and h(t) < λ < ν. Letting ν tend to f (t), obtain h(t) ≤ f (t). The same inequality is immediate for f (t) = +∞. By analogy, Vµ ⊂ Uλ for µ < λ. Hence, f (t) ≤ g(t) for all t ∈ Q. Writing (b) as Wµ ⊂ Vλ (µ < λ), and arguing as above, conclude that g(t) ≤ h(t) for all t ∈ Q. Therefore, f = g = h. The fact that f is continuous follows from the equalities [ {Vµ : µ < λ, µ ∈ ƒ}, \ {f ≤ λ} = {h ≤ λ} = {Wµ : µ > λ, µ ∈ ƒ}, {f < λ} = {g < λ} =

since Vµ is open whereas Wµ is closed for all µ ∈ ƒ. B (2) Let Q be a quasiextremal compact space. Assume that Q0 is an open dense Fσ -set in Q and f : Q0 → R is a continuous function. Then there is a unique continuous function f¯ : Q0 → R such that f (t) = f¯(t) (t ∈ Q0 ). C Indeed, if Uµ := cl({f < µ}) then the mapping µ 7→ Uµ (µ ∈ R) increases and meets the condition (b) of (1). Therefore, there is a unique function f¯ : Q → R satisfying {f¯ < µ} ⊂ Uµ ⊂ {f¯ ≤ µ} (µ ∈ R). Obviously, in this case f¯  Q0 = f , i.e. the restriction of f¯ to Q0 coincides with f . B 1.4.2. Let Q be a quasiextremal compact space. Denote by C∞ (Q) the set of all continuous functions x : Q → R assuming the values ±∞ possibly on a nowheredense set. Order C∞ (Q) by assigning x ≤ y whenever x(t) ≤ y(t) for all t ∈ Q. Then, take x, y ∈ C∞ (Q) and put Q0 := {|x| < +∞} ∩ {|y| < +∞}. In this case Q0 is open and dense in Q. According to 1.4.1 (2), there is a unique continuous function z : Q → R such that z(t) = x(t) + y(t) for t ∈ Q0 . It is this function z that we declare the sum of x and y. In an analogous way we define the product of a pair of elements. Identifying the number λ with the identically λ function on Q, we obtain the product of x ∈ C∞ (Q) and λ ∈ R.

Boolean Algebras and Vector Lattices

27

Clearly, the space C∞ (Q) with the operations and order introduced above is a vector lattice and a faithful f -algebra. The identically one function 1 is a ring and order-unity. The order ideal generated by 1 is the space C(Q) of all continuous numeric functions on Q. (1) The space C∞ (Q) is a universally σ-complete Kσ -space. T∞ C Take a bounded increasing sequence (xn ) of elements of C∞ (Q). Put Vλ := n=1 {xn ≤ λ} and Uλ := int Vλ . Then Vλ is a closed Gδ -set and, by assumption, Uλ is a clopen set. According to 1.4.1 (2) there is a unique function x : Q → R such that {x < λ} ⊂ Uλ ⊂ {x ≤ λ} for all λ ∈ R. Now, it is not difficult to check that x = supn xn . Universally σ-completeness is obvious. B (2) The base of the vector lattice C∞ (Q) is isomorphic to the Boolean algebra of all regular open (closed) subsets of Q. C The same argument as used in (1) works. B (3) The space C∞ (Q) is an order complete vector lattice if and only if Q is extremal. C In the case when Q is extremal, the order completeness of C∞ (Q) may be proved as in (1). The converse follows from the Ogasawara Theorem 1.2.4, since Boolean algebras Clop(Q) and E(1) are isomorphic. B According to these arguments we may describe suprema and infima in C∞ (Q) as follows. If (xα ) is an order-bounded family in C∞ (Q) then x = supα xα if and only if there exists a comeager subset Q0 ⊂ Q such that x(t) = supα xα (t) for all t ∈ Q0 . 1.4.3. According to 1.3.8, to each element of a Kσ -space with order-unity there corresponds its spectral function; moreover, the operations transform in a rather definite way. This circumstance suggests that an arbitrary Kσ -space with unity can be realized as a space of “abstract spectral functions.” We will expatiate upon this. A resolution of unity or resolution of the identity in a Boolean algebra B is defined as a mapping e : R → B satisfying the conditions (1) s ≤ t → e(s) ≤ e(t) (s, t ∈ R); W V (2) t∈R e(t) = 1, t∈R e(t) = 0; W (3) s∈R,s
(e0 , e00 ∈ K(B)).

Further, suppose that B is a σ-algebra and choose any dense countable subfield P of R. By property (3), every resolution of unity is uniquely determined by its values on P.

28

Chapter 1 Given e0 , e00 ∈ K(B), we may define the mapping W e : t 7→ {e0 (r) ∧ e00 (s) : r, s ∈ P, r + s = t} (t ∈ P), W e : t 7→ {e(s) : s ∈ P, s < t} (t ∈ R)

which is obviously a resolution of unity in B. Putting e0 + e00 := W e, we obtain the structure of a commutative group in K(B); moreover, −e(t) = {1 − e(−s) : s ∈ ¯ have the form: 0(t) ¯ := 1 if t > 0 and 0(t) ¯ := 0 if P, s < t} and the zero element 0 ¯ := 1 if t > 1 and 1(t) ¯ := 0 if t ≤ 1. Finally, define the product of t ≤ 0. Set 1(t) an element e ∈ K(B) and a real α ∈ R by the rules (αe)(t) := e(t/α)

(α > 0, t ∈ R),

(αe)(t) := (−e)(−t/α)

(α < 0, t ∈ R).

To each element b ∈ B assign the resolution of unity ¯b defined as ¯b(t) := 1 if t > 1, ¯b(t) := b∗ := 1 − b if 0 ≤ t < 1, and ¯b(t) := 0 if t ≤ 0. Theorem. Let B be a complete (σ-complete) Boolean algebra. The set K(B) with the above-introduced operations and order represents a universally complete ¯ serves as an orderK-space (a universally σ-complete Kσ -space). The element 1 ¯ unity and the mapping b 7→ b (b ∈ B) is an isomorphism of Boolean algebras B and ¯ E(1). C The proof is elementary. Most effort is put into the routine calculation with resolutions of unity. B 1.4.4. Theorem. Let E be a Kσ -space with order-unity 1 and B := E(1). The mapping sending an element x ∈ E to the spectral function λ 7→ exλ (λ ∈ R) is an isomorphism of E onto an order-dense ideal in K(B). If E is universally σ-complete then E and K(B) are isomorphic. C Denote by h the mapping from E to K(B) we are interested in. By virtue of Theorem 1.3.8 h is linear and preserves order. Moreover, h is injective according to the Freudenthal Spectral Theorem 1.3.9. Prove that h(E) is an order ideal in K(B). Assume 0 ≤ s ≤ h(y) where s ∈ K(B) and y ∈ E. Let λ > µ, b := s(λ) − s(µ), and y0 := µb. It can be easily checked that µ¯b ≤ ¯bs, since  if t > µ,    1, b ∧ s(t) + b∗ , if t > 0, ∗ (µ¯b)(t) = b , if 0 < t ≤ µ, (¯bs)(t) =  0, if t ≤ 0.  0, if t ≤ 0; Therefore, h(y0 ) = µh(b) = µ¯b ≤ ¯bs ≤ s ≤ h(y) and we obtain y0 ≤ y. Finally, we deduce λ(s(λ) − s(µ)) = y0 + (λ − µ)b ≤ y + (λ − µ)1.

Boolean Algebras and Vector Lattices

29

Now, consider a partition of the real axis of the form βN := (tn )n∈Z where tn := n/N and N ∈ N. The disjoint sum ¯(βN ) := x

X

tn+1 (s(tn+1 ) − s(tn ))

n∈Z

exists in E, since tn+1 (s(tn+1 ) − s(tn )) ≤ y + (1/N )1 as was proven above. Denote ¯(βN ). Every element of the form by A the sequence of all elements x x(βN ) :=

X

tn (s(tn+1 ) − s(tn ))

n∈Z

x(β)}. Note that is a lower bound of A. Therefore, there exists x := inf(A) := inf{¯ x ¯(β)

eλ

=

W

{s(tn ) : tn < λ}.

Hence, by 1.3.8 (8), we infer exλ =

_ a∈A

eaλ =

_

s(t) = s(λ) (λ ∈ R).

t∈R,t<λ

Thereby h(x) = s and h(E) is an order ideal. Moreover, it can be easily observed that for a resolution of unity 0 ≤ s ∈ K(B) the following equivalences are true: s ⊥ h(E) ⇔ (∀b ∈ B) s ⊥ h(b) ⇔ (∀b ∈ B) s ⊥ ¯b ⇔ (∀b ∈ B) (∀t ≥ 0) s(t) ∨ ¯b(t) = 1 ⇔ (∀b ∈ B) (∀0 ≤ t < 1) s(t) ∨ b∗ = 1 ⇔ (∀b ∈ B) (∀0 ≤ t < 1) s(t) ≥ b ⇔ (∀0 ≤ t < 1)s(t) = 1 ⇔ s = 0. This proves that h(E) is order dense in K(B). Thus, it remains to verify that h is surjective in the case when E is universally σ-complete. For an arbitrary resolution of unity s in the Boolean algebra B the disjoint sum X ¯(βN ) := x tn+1 (s(tn+1 ) − s(tn )) n∈Z

exists in the universally σ-complete Kσ -space E. The same argument as above applies, and we again arrive at h(x) = s with x := inf{¯ x(β)}. B 1.4.5. Theorem. Suppose that E is a universally σ-complete Kσ -space with π is the order-unity 1, Q is the Stone space of the Boolean algebra E(1) and π 7→ ^ canonical isomorphism from E(1) onto Clop(Q). Then E is linearly and latticially

30

Chapter 1

isomorphic to C∞ (Q). Moreover, such an isomorphism may be carried out by sending x ∈ E to x b : Q → R by the rule x b(q) := inf{λ ∈ R : exλ ∈ q}. C As was shown in 1.4.4, the Kσ -space E is isomorphic to the space of all B-valued spectral functions, with the function λ 7→ exλ (λ ∈ R) corresponding to x ∈ E. Assume that a clopen sunset Uλ of the Stone space Q corresponds to exλ ∈ B. Then, by virtue of 1.4.1 (2), to every element x ∈ E there corresponds a unique continuous function x b : Q → R such that {b x < λ} ⊂ Uλ ⊂ {b x ≤ λ}. In this case, however, x b(q) = inf{λ ∈ R : q ∈ Uλ } = inf{λ ∈ R : exλ ∈ q}. V x W x The formulas {e } = 0 and {eλ } = 1 (cf. 1.3.6 (2))S imply that the interior of the λ T closed set {Uλ : λ ∈ R} is empty while the open set {Uλ : λ ∈ R} is dense in Q. Therefore, the function x b may assume the values ±∞ only on a nowhere-dense set and so x b ∈ C∞ (Q). We omit the elementary demonstration of the fact that x 7→ x b is a linear and lattice isomorphism. B 1.4.6. We now list a few corollaries to the above theorems. (1) Let E be an arbitrary K-space. Assume further that {eξ }ξ∈„ is a total disjoint positive family in E. Denote by Q the Stone space of the Boolean algebra of bands of B(X). Then there is a unique linear and lattice isomorphism of E onto an order-dense ideal of the K-space C∞ (Q) such that eξ transforms into the characteristic function of a clopen subset Qξ of Q. This isomorphism sends x ∈ E to the function x b : Q → R acting by the rule  x b(q) := inf λ ∈ R : {eξλ }⊥⊥ ∈ q (q ∈ Qξ ), where λ 7→ (eξλ ) is the spectral function of the band projection of x to {eξ }⊥⊥ with respect to the order-unity eξ . (2) A space E is a universally complete K-space (K-space of bounded elements) if and only if the image of E under the above isomorphism is the whole of C∞ (Q) (the subspace C(Q) of all continuous functions on Q). (3) Every universally complete K-space E can be endowed with a multiplication making E into a faithful commutative f -algebra. Moreover, if we require the order-unity fixed in E to be the multiplication unity, then this multiplication in E is unique. (4) Let D be a subset of a K-space E with an upper bound e ∈ E. Then inf D = 0 if and only if for every o < ε ∈ R there exists a partition of unity (πξ )ξ∈„ in P(E) and a family (dξ )ξ∈„ in D such that πξ dξ ≤ εe for all ξ ∈ „.

Boolean Algebras and Vector Lattices

31

C Without loss of generality we may assume that {e}⊥⊥ = E = C∞ (Q). If inf D = 0 then, in view of 1.4.2, there is a comeager subset Q0 ⊂ Q such
that inf{d(t) : d ∈ D} = 0 for all t ∈ Q0 . Put Rd := cl {t ∈ Q : d(t) < εe(t)} . Then Rd S is a clopen set and the least upper bound of the family (Rd )d∈Q0 in Clop(Q) is Q ( d∈D Rd contains Q0 and is dense in Q). By the Exhaustion Principle there is a partition of unity (Qξ )ξ∈„ in Clop(Q) with the property: for each ξ ∈ „ there is some dξ ∈ D such that Qξ ⊂ Rdξ . Take the band projection corresponding to Qξ as πξ and we are done. The converse is evident. B 1.4.7. (1) Let (Š, A ) be a measurable space, i.e., Š is a nonempty set and A is a σ-algebra of its subsets. Denote by M (Š, A ) the set of all real (complex) measurable functions on Š and furnish M (Š, A ) with the pointwise operations and order induced from RŠ (from C Š , with C the complex numbers). Choose some σcomplete ideal N of the algebra A . Let N comprise the functions f ∈ M (Š, A ) such that {t ∈ Š : f (t) 6= 0} ∈ N . Assign M (Š, A , N ) := M (Š, A )/N . Then M (Š, A ) and M (Š, A , N ) are real (complex) Kσ -spaces and f -algebras at the same time. (2) Consider an arbitrary measure space (Š, †, µ), see 1.1.7 (5). Denote L (Š, A , µ) := M (Š, A , µ) := M (Š, A , µ−1 (0)). Again, L0 (Š, A , µ) is a vector lattice. The coset 1 of the identically one function on Š is an order and ring unity in M (Š, A , µ). Denote by L∞ (Š, A , µ) the principal ideal in L0 (Š, A , µ) generated by 1. Let L ∞ (Š) := L ∞ (Š, A , µ) stand for the part of M (Š, A comprising all essentially bounded functions. If µ is a finite or σ-finite measure then L∞ (Š, A , µ) and L0 (Š, A , µ) are order complete; moreover L0 (Š, A , µ) is also universally complete. In general, the order completeness of L0 (Š, A , µ) relates to the direct sum property for µ (see 1.1.7 (5) and 1.4.8 below). 0

(3) The space L0 (Š, A , µ) is diffuse if and only if µ has no atoms. Recall that an atom of a measure µ is a set A ∈ A such that µ(A) > 0 and A0 ∈ A , A0 ⊂ A, implies either µ(A0 ) = 0 or µ(A0 ) = µ(A). The discreteness of L0 (Š, A , µ) is equivalent to the fact that the measure µ is purely atomic, i.e., every set of nonzero measure contains an atom of µ. The equivalence class containing the identically unity function is an order and ring unity in L0 (Š, A , µ). 1.4.8. A mapping ρ : L∞ (Š) → L ∞ (Š) is said to be a lifting of L∞ (Š) if for all α, β ∈ R and f, g ∈ L∞ (Š) the following are true: (a) ρ(f ) ∈ f and dom(ρ(f )) = Š; (b) if f 6 g, then ρ(f ) 6 ρ(g) everywhere on Š; (c) ρ(αf +βg) = αρ(f )+βρ(g), ρ(f g) = ρ(f )ρ(g), ρ(f ∨g) = ρ(f )∨ρ(g), ρ(f ∧ g) = ρ(f ) ∧ ρ(g);

32

Chapter 1

(d) ρ(0) = 0 and ρ(1) = 1 everywhere on Š. Theorem. For a measure space (Š, A , µ), the following are equivalent: (1) (Š, A , µ) possesses the direct sum property; (2) L∞ (Š) admits a lifting; (3) L0 (Š, A , µ) is order complete and universally complete. The base of the K-space L0 (Š, A , µ) is isomorphic to the Boolean algebra B(Š, A , µ) of measurable sets modulo zero-measure sets. 1.4.9. Suppose that (Š, A , µ) is a measure space with a direct sum property. Let ρ be a lifting of L∞ (Š, A , µ). and τ : Š → Q be the corresponding canonical immersion of Š into the Stone space Q of the Boolean algebra B(Š, A , µ), see Denote by τ ∗ the mapping that sends each function f ∈ C∞ (Q) to the equivalence class of the measurable function f ◦ τ . Theorem. The mapping τ ∗ is a linear and order isomorphism from C∞ (Q) onto L0 (Š, A , µ). The image of C(Q) under the isomorphism τ ∗ coincides with L∞ (Š, A , µ). For u ∈ L0 (Š, A , µ) the function u b := τ ∗−1 (u) is called the Stone transform of u. 1.4.10. Now, we will answer the question: which vector lattices are representable as order-dense ideals in L0 (Š, A , µ)? An order complete vector lattice E is said to be a Kantorovich–Pinsker space if its base B(E) is a multinormed Boolean algebra (= there exists an essentially positive locally finite countably additive measure on B(E), see 1.2.9), or equivalently, if E contains an order-dense ideal with a total set of order continuous functionals. It can be proved that if (Š, A , µ) is a measure space possessing the direct sum property then L0 (Š, A , µ) is a Kantorovich–Pinsker space. Theorem. Each Kantorovich–Pinsker space E is linearly and order isomorphic to an order-dense ideal of L0 (Š) for a suitable measure space (Š, A , µ) with the direct sum property. If an order-unity 1 is fixed in E, then among such isomorphisms there is a unique isomorphism taking 1 to the equivalence class of the identically one function on Š. The space E is universally complete if and only if the image of each isomorphism from E onto an order-dense ideal of L0 (Š) coincides with L0 (Š). C The proof can be obtained by combining 1.2.10, 1.4.5, and 1.4.9. B 1.4.11. Let (Š, A , µ) be a measure space with the direct sum property. An arbitrary order ideal E in L0 (Š, †, µ) is called an ideal space on (Š, †, µ). We recall some basic properties of ideal function spaces. In this subsection fe := f ∼ stands for the coset of a measurable function f . (1) Every ideal space is an order complete vector lattice.

Boolean Algebras and Vector Lattices

33

(2) Let M be a subset in L0 (Š, A , µ) unbounded from above. Then M contains a countable subset that is also unbounded from above. (3) If µ is σ-finite then every nonempty order-bounded set M in ideal space contains a countable subset (fen ) with sup(M ) = supn fen . Moreover, fe := supn fen can be computed pointwise: f (t) = supn fn (t) (t ∈ Š). (4) A sequence (fen ) ⊂ E o-converges to fe if and only if it is orderbounded (in E) and fn (t) → f (t) for almost all t ∈ Š. For fe ∈ L0 (Š, A , µ) denote the support of fe by supp fe:= supp(f ) := {t ∈ Š : f (t) 6= 0}. Let ef stand for the coset of the characteristic function of supp(f ). The support of a nonempty subset M ⊂ L0 (Š, A , µ) is a measurable subset Š0 ⊂ Š such that sup{ef : f ∈ M } is the coset of the characteristic function of Š0 . Observe that supp(M ) is determined up to a set of measure zero. We write A ⊂ B mod (µ) if µ(B \ A) = 0. (5) Let M be a nonempty subset of E, Š0 := supp(M ) and Š1 := Š\Š0 . Then n o ⊥ e M = f ∈ E : supp(f ) ⊂ Š1 mod (µ) , n o M ⊥⊥ = fe ∈ E : supp(f ) ⊂ Š0 mod (µ) . (6) Let K be a band in E, and let χK be the characteristic function of supp(K). Then the band projection [K] has the form: [K](fe) = (χK f )∼ (fe ∈ E). (7) An ideal space E is order dense in L0 (Š, A , µ) if and only if supp(E) = Š mod (µ). If, in addition, µ is σ-finite then for every 0 ≤ f ∈ L0 (Š, A , µ) there exists an increasing sequence (fn ) in E such that f = o-limn fn . 1.5. Normed Vector Lattices In this section we expose various classes of normed vector lattices determined by the interplay of order and norm. Some fundamental properties are given. A more detailed presentation can be found in [23, 162, 231, 242, 336, 341, 388]. 1.5.1. Let E be a vector lattice. A norm k · k on E is called a lattice norm if |x| ≤ |y| implies kxk ≤ kyk for all x, y ∈ E. A lattice norm can be equivalently defined by following two relations: 0 ≤ x ≤ y ⇒ kxk ≤ kyk (x, y ∈ E) and k |x| k = kxk (x ∈ E). If k · k is a lattice norm on E, the pair (E, k · k) is called a normed (vector) lattice. A norm complete normed lattice is called a Banach lattice. It is immediate from 1.3.2 (1, 8) and monotonicity of a lattice norm that the mappings x 7→ x+ , x 7→ x− , x 7→ |x|, and the lattice operations (x, y) 7→ x ∨ y

34

Chapter 1

and (x, y) 7→ x ∧ y are uniformly continuous. In particular, the positive cone, the disjointness relation, and every band are closed. The closure of a vector sublattice (order ideal) of E is a vector sublattice (order ideal) of E. It is also clear that each of the lattice operations of a normed lattice E admits a unique continuous e so that E e becomes a Banach lattice. We state extension to the norm completion E also a useful criterion for norm completeness. For an arbitrary normed vector lattice E the following are equivalent: (1) E is a Banach lattice; (2) each increasing Cauchy sequence of positive elements in E has the least upper bound; (3) every absolutely convergent series of positive elements in E is order convergent. If, in addition, E has the principal projection property then each of the following two conditions is equivalent to (1): (4) each laterally increasing Cauchy sequence of positive elements in E has the least upper bound; (5) every absolutely convergent series of pairwise disjoint positive elements in E is order convergent. 1.5.2. Consider an arbitrary vector lattice E. A linear functional f on E is called order-bounded if the image of every order interval in E under f is a bounded set in R. Denote by E ∼ the set of all order-bounded functionals on E. A functional f is positive if f (x) ≥ 0 for all x ∈ E+ . The space E ∼ becomes an ordered vector ∼ of all positive functionals and space if a vector order in it is defined by the cone E+ is referred to as order dual space. A linear functional f ∈ E ∼ is called order continuous if limα f (xα ) = 0 for every decreasing net (xα ) in E with inf xα = 0. The set of all order continuous functionals is denoted by En∼ . In the sequel E 0 stands for the space of all norm continuous functionals on a normed lattice E. Each x ∈ E defines by x ^ : f 7→ hx, f i = f (x) (f ∈ E ∼ ) an order-bounded ∼ functional on E . The mapping assigning x ^ to each x ∈ E by hf, x ^i = hx, f i (f ∈ ∼ E ) is called the canonical embedding in the second order dual or the evaluation mapping This mapping preserves order and is injective provided that E ∼ is pointseparating. We now list a few basic properties of order-bounded functionals and order duals. (1) The order dual space is an order complete vector lattice. (2) The set En∼ is a band in E ∼ . (3) The evaluation mapping is an isomorphism from E onto a sublattice

Boolean Algebras and Vector Lattices

35

of E ∼∼ provided that E ∼ is point-separating. (4) Let E0 be a massive subspace of an ordered vector space E. Each positive linear functional on E0 has a positive linear extension to E. (5) If E is a normed lattice then E 0 is an order ideal in E ∼ . (6) If E is a normed lattice then E 0 is an order complete Banach lattice. (7) If E is a Banach lattice then E 0 = E ∼ and En∼ is a band in E 0 . (8) Let E0 be a vector sublattice of the normed vector lattice E. Each norm continuous positive linear functional on E0 has a norm preserving positive linear extension to E. 1.5.3. A Banach lattice is said to have an order continuous norm (σ-order continuous norm) if limα kxα k = 0 for every downward directed net (sequence) (xα ) with inf α xα = 0. In the Russian literature, order continuity is frequently called Property (A). We say that the norm in E is laterally σ-continuous if limn ken k = 0 for every decreasing sequence (en ) in E such that (en − en+1 ) ⊥ en+1 (n ∈ N) and inf n en = 0. Theorem. For every Banach lattice E, the following are equivalent: (1) The norm of E is order continuous; (2) E is order complete and the norm of E is order continuous; (3) E is order σ-complete and the norm of E is σ-order continuous; (4) E is order complete and the norm of E is laterally σ-continuous; (5) E is order σ-complete and each closed order ideal of E is a band; (6) every closed order ideal of E is a projection band; (7) every order interval of E is σ(E, E 0 )-compact; (8) every norm continuous linear functional on E is order continuous; (9) the canonical embedding E → E 00 maps E onto an ideal of the Banach lattice E 00 . C The proof can be found in [242, 263, 336, 341]. B 1.5.4. We say that a normed lattice E is monotonically complete or possesses Property (B) if for every increasing net (xα ) in E+ with supα kxα k < ∞ there exists the supremum x = supα xα . Every monotonically complete normed lattice is order complete. A vector lattice E is said to have the order semicontinuous norm or to possess Property (C) if o-limα xα = x implies limα kxα k = kxk for every increasing net (xα ) in E+ . The sequential variants of Properties (A), (B), and (C) are denoted by (A)σ , (B)σ , and (C)σ , respectively. A Banach lattice possessing (A)σ and (B)σ is called a KB-space.

36

Chapter 1

The significance of these properties for the theory of Banach lattices has various aspects. We emphasize only those related to the properties of canonical embedding into the second dual lattice. (1) A Banach lattice E with En∼ point-separating is monotonically complete if and only if the canonical embedding E → (En∼ )∼ n is an isometry. (2) An order complete Banach lattice E with En∼ point-separating has order semicontinuous norm if and only if the canonical embedding E and (En∼ )∼ n are isomorphic under the canonical embedding. (3) Let E be an order complete Banach lattice with En∼ point-separating. Then E is monotonically complete and has the order semicontinuous norm if and only if the image of E under the canonical embedding is the range of a positive contractive projection. (4) A Banach lattice E is a KB-space if and only if the canonical embedding κ maps E onto a band in E 00 . 1.5.5. (1) A Banach lattice E is called an abstract M -space or AM -space, for short, if kx ∨ yk = kxk ∨ kyk (x, y ∈ E+ ). If the unit ball of an AM -space E contains a largest element e, then e is a strong order-unity and the unit ball of E coincides with the symmetric order interval [−e, e]. In this case E is said to be an AM -space with unity. (2) Let E be an arbitrary vector lattice and u ∈ E. We may introduce the following seminorm in the principal ideal E(u): kxku := inf{λ ∈ R : |x| ≤ λu} (x ∈ E(u)). The seminorm k · ku is a norm if and only if the lattice E(u) is Archimedean. It can be easily seen that relative uniform convergence in E is the convergence in the norm of (E(v), k · kv ). Therefore, a vector lattice E is relatively uniformly complete if and only if the normed lattice (E(u), k · ku ) is complete for every u ∈ E+ . Let E be a Banach space and 0 6= u ∈ E. Then (E(u), k · ku ) is an AM -space with unity u and the identity embedding E(u) → E is continuous. This simple proposition asserts that every Banach lattice is locally arranged as an AM -space, which enables us to reduce the study of some properties of Banach lattices to that of AM -spaces. Such an approach is demonstrated below in 1.5.7, 1.5.11, and 1.5.12. (3) A Banach lattice E is called an ALp -space if kx + yk = kxkp + kykp


p1

Boolean Algebras and Vector Lattices

37

for all disjoint x, y ∈ E+ . Here 1 ≤ p ≤ ∞ and, in case p := ∞, we let (tp + sp )1/p := max{s, t} (0 < s, t ∈ R). The terms AM -space and AL-space are conventionally used instead of AL∞ -space and AL1 -space, respectively. This definition of AM space is equivalent to that given in (1) despite the fact that the required equality holds only for disjoint pairs of elements. 1.5.6. We recall three well-known facts from the theory of Banach lattices. Proofs can be found in [231, 242, 341]. (1) Theorem. Let E be an AM -space. Then there exist a compact space Q and a family of triples (tα , sα , λα )α∈A with tα , sα ∈ Q and 0 ≤ λα < 1 such that E is isometrically isomorphic to the closed sublattice F := {x ∈ C(Q) : (∀α ∈ A) x(tα ) = λα x(sα )}. (2) Brothers Kre˘ın–Kakutani Theorem. Every AM -space with unity is linearly isometric and order isomorphic to the space of continuous functions C(Q) on some compact space Q. (3) Theorem. If 1 ≤ p < +∞ then every ALp -space is linearly isometric and order isomorphic to Lp (Š, A , µ) for a suitable measure space (Š, A , µ) with the direct sum property. 1.5.7. The study of functions f : Rl → R, for which f (e1 , . . . , el ) can naturally be defined for e1 , . . . , el ∈ E, is called the functional calculus. To assign some values to expressions of the form f (e1 , . . . , el ) we will use the local AM -structure of a Banach lattice. Denote by H (Rl ) the space of all real continuous functions f on Rl which are positively homogeneous, i.e. f (λt) = λf (t) for all t ∈ Rn and λ ≥ 0. Clearly, H (Rn ) is a vector lattice under pointwise operations. Denote S := {(t1 , . . . , tl ) ∈ Rl : |t1 | + · · · + |tl | = 1}. Clearly, each function f ∈ H (Rl ) is completely defined by its values on S. Therefore, the restriction mapping f 7→ f |S is a linear and lattice isomorphism from H (Rl ) onto C(S). Thus, we may regard H (Rl ) as a Banach lattice with strong order-unity and norm kf k∞ := sup{|f (x)| : x ∈ S}. We define dxj ∈ H (Rn ) by dxj (t1 , . . . , tn ) = tj (j := 1, . . . , n). Theorem. Let E be a uniformly complete vector lattice, e1 , . . . , el ∈ E, and e := e1 + · · · + en . Then there is a unique lattice homomorphism h : H (Rl ) → E
l such that h(dxj ) = ej (j := 1, . . . , l). Moreover, h H (R ) is the e-uniform closure of the vector sublattice generated by {e1 , . . . , el }. C See [242, 341]. B 1.5.8. A normed (Banach) ideal space on (Š, †, µ) is defined as an ideal space E on (Š, †, µ) endowed with a lattice norm making E into a normed (Banach) space.

38

Chapter 1

If a sequence xn converges to x in the norm of a Banach ideal space E then xn → x(µ). If (xn ) is a Cauchy sequence in E then it converges in measure to some x ∈ L0 (T, †, µ). Now, we consider some properties of the vector lattice C(Q) of all continuous functions on a compact topological space Q. A vector lattice is called disjointly complete (disjointly σ-complete if every its order-bounded antichain (countable antichain) has supremum. 1.5.9. Theorem. For a compact space Q, the following are equivalent: (1) C(Q) is order complete (σ-complete); (2) C(Q) is disjointly complete (σ-complete); (3) Q is extremal (quasiextremal); (4) C(Q) possesses the projection property (principal projection property). C The implications (1) ⇒ (2) and (1) ⇒ (4) areSobvious. ∞ (2) ⇒ (3): Take an open G ⊂ Q such that G = n=1 Fn with Fn closed for all n ∈ N. We will prove that cl G is open. Since every compact topological space is normal, we may find an open G1 ⊂ Q with F1 ⊂ G1 ⊂ cl G1 ⊂ G. By the same argument there is an open G2 ⊂ Q with cl G1 ∪ F2 ⊂ G2 ⊂ cl G2 ⊂ G. By induction we may construct a sequence (Gn ) of open subsets such that cl Gn ∪Fn+1 ⊂ Gn+1 ⊂ cl Gn+1 ⊂ G. Clearly, ∞ ∞ [ [ G= Gn = cl Gn . n=1

n=1

Without loss of generality, we may assume that all Gn differ from one another. Now, we define a sequence of continuous functions (xn ) such that 0 ≤ xn ≤ 1 and  1, if t ∈ cl Gn−1 , xn (t) := 0, if t ∈ cl Gn−2 ∪ (Q \ Gn−1 ), where G−1 := G0 := ∅. It is easily seen that each of the sequences (x3n ), (x3n−1 ), and (x3n−2 ) is pairwise disjoint. By assumption there exist yk := supn∈N x3n−k (k := 0, 1, 2). Put y := y0 ∨ y1 ∨ y2 . Obviously, yk (Q \ cl G) = {0} (k := 0, 1, 2), and therefore y(Q \ cl G) = {0}. At the same time y(G) = {1}. Since y is continuous, we have y(cl G) = {1}. Thus, y = χcl G and cl G is clopen set. Now, suppose that C(Q) is disjointly complete. Then Q is quasiextremal as we have just proved. Moreover, the Boolean algebras Clop(Q) and E(1) are isomorphic; therefore, Q is the Stone space of the Boolean algebra of clopen sets Clop(Q). Whence the required assertion follows from 1.1.6 (2) and the Ogasawara Theorem (1.2.4).

Boolean Algebras and Vector Lattices

39

(3) ⇒ (1): follows from 1.4.2 (1, 3). (4) ⇒ (3): See [341]. B 1.5.10. Theorem. Let Q be a compact space and E be a vector sublattice in C(Q) containing constants and separating the points of Q. If E is disjointly complete then Q is extremal and E possesses the projection property. C First, we prove that Q is extremal. In view of 1.5.9 it suffices to establish disjoint completeness of C(Q). Take a pairwise disjoint family (xξ )ξ∈„ , 0 ≤ xξ ≤ 1, and put  xξ (t), if xξ (t) > 0, z(t) := 0, if t ∈ G := int{t ∈ Q : (∀ξ ∈ „)xξ (t) = 0}. Clearly, z is a continuous function defined on an open dense subset in Q. According to the Stone–Weierstrass Theorem, for each ξ ∈ „ there exists a sequence (yξ,n ∈ E+ ), such that 0 ≤ xξ − yξ,n ≤ (1/n)1, (n ∈ N). By assumption there exists yn := supξ∈„ yξ,n . This sequence is uniformly convergent since |yn+p − yn | ≤ (1/n)1 (n ∈ N). Put x := lim yn . Observe that 0 ≤ z(t)−yn (t) ≤ (1/n)1 for all t ∈ dom(z); thus x(t) = lim yn (t) = z(t) for all t ∈ dom(z). This means that x is a continuous extension of z onto Q; therefore x = supξ∈„ xξ . Now, take an arbitrary element e ∈ E+ and prove that eχQ0 ∈ E for any clopen set Q0 ⊂ Q. Since E contains constants and separates points of Q, for each q ∈ Q0 there exists eq ∈ E with eq (q) > e(q). Put Uq := {t ∈ Q : eq (t) > e(t)} and choose a finite subcover Uq1 , . . . , Uqm of Q0 . Then the following relations are consistent: e0 := eq1 ∨ · · · ∨ eqm ∈ E; e0 (t) = 0 (t ∈ Q \ Q0 );

e0 (t) ≥ e(t) (t ∈ Q0 ).

Now it is clear that e ∧ e0 = eχQ0 ∈ E. B 1.5.11. Theorem. Every Archimedean disjointly complete vector lattice has the projection property. Every Archimedean disjointly σ-complete vector lattice has the principal projection property. C We consider the claim only for disjointly complete vector lattices. The disjointly σ-complete case is handled by the same arguments. Disjoint completeness of E is inherited by order ideals. In particular, all principal ideals E(u) are disjointly ] of E(u) is an AM -space. Using complete. Clearly, the norm completion E(u) ] is isomorphic to C(Q) for some compact space Q. 1.5.6 (2) we conclude that E(u) The isomorphism brings E(u) in a vector sublattice containing constants and separating points of Q. According to 1.5.10 E possesses the projection property. B

40

Chapter 1

1.5.12. Theorem. Let E be a Banach lattice or relatively uniformly complete vector lattice. Then the following are equivalent: (1) E is order complete (order σ-complete); (2) E is disjointly complete (disjointly σ-complete); (3) E has the projection property (principal projection property). C The implication (1) ⇒ (2) is obvious and (2) ⇒ (3) follows from 1.5.11. To prove (3) ⇒ (1) use the local AM -structure of a Banach lattice (relatively uniformly complete vector lattice) and apply (1) ⇔ (4) from 1.5.9. B 1.5.13. Let A be a Boolean algebra. Denote by ba(A ) the set of all bounded finitely additive functions µ : A → R with a vector structure induced from RA . The set ba+ (A ) of all positive bounded finitely additive functions is a cone and defines some structure of a vector lattice in ba(A ). The lattice operations have the form: µ ∨ ν(a) = sup{µ(b) + µ(a ∧ b∗ ) : b ≤ a}, µ ∧ ν(a) = inf{µ(b) + µ(a ∧ b∗ ) : b ≤ a}, |µ|(a) = sup{|µ(b)| : b ≤ a}, where µ, ν ∈ ba(A ). The total variation of µ is defined as kµk := |µ|(1) where 1 is the unity of A . It is clear that the function µ 7→ kµk is a lattice norm. The vector lattice ba(A ) of bounded, finitely additive functions on a Boolean algebra is an AL-space. In particular, ba(A ) is order complete and the norm of ba(A ) is order continuous. 1.6. Comments 1.6.1. (1) The theory of Boolean algebras originated from the classical work by G. Boole “An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities” [49, 50]. The author himself formulated his intentions as follows: “The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the language of a Calculus, and upon this foundation to establish the science of Logic and construct its method to make that method itself the basis of a general method for the application of the mathematical doctrine of probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.... ” Pursuing this end, G. Boole carried out, in fact, the algebraization of the logical system lying behind the classical mathematical reasoning. In a result, he become the author of the algebraic system omnipresent under the name of Boolean algebra.

Boolean Algebras and Vector Lattices

41

(2) Definition 1.1.2 looks somewhat strange at first sight. Indeed, it does not reveal the reasons for whatever distributive lattice to be called an algebra since the term “algebra” refers customarily to conventional objects (cf. Lie algebra, Banach algebra, C ∗ -algebra, etc.). The arising ambiguity is easily eliminated because a Boolean algebra is in fact an algebra over the two-element field. At the same time, it is perfectly natural to view Boolean algebras in different contexts at different angles. It is worth emphasizing that the particular Boolean algebras we deal with in functional analysis appear mostly as distributive complemented lattices. (3) Recall that a universal algebra is an algebraic system without predicates. This concept makes available another definition of Boolean algebra. Namely, a Boolean algebra B is a universal algebra (B, ∨, ∧, ∗, 0, 1) with two binary operations ∨ and ∧, one unary operation ∗, and two distinguished elements 0 and 1 obeying the conditions: (a) ∨ and ∧ are commutative and associative; (b) ∨ and ∧ are both distributive relative to one another; (c) x and x∗ complement one another; (d) 0 and 1 are neutral for ∨ and ∧, respectively. Conversely, given a universal algebra B of the above type, make B into a poset by letting x ≤ y whenever x ∧ y = x for x, y ∈ B. In this event, note that (B, ≤) is a distributive complemented lattice with join ∨, meet ∧, complementation ∗, zero 0, and unity 1. 1.6.2. (1) Theorem 1.2.3 shows that every Boolean algebra is completely determined from its Stone space. In more detail, each property of a Boolean algebra B translates into the topological language, becoming a property of the Stone space S (B) of B. This way of studying Boolean algebras is the Stone representation method. A detailed presentation can be found in [144, 283, 352]. (2) Denote by Bool the category of Boolean algebras and Boolean homomorphisms, and let Comp stand for the category of compact spaces and continuous mappings. Then 1.2.3 and 1.2.5 may be paraphrased as follows: Theorem. The mapping S is a contravariant functor from the category Bool to the category Comp. (3) About measure spaces with the direct sum property see [78, 162, 239, 375]. The problem of whether lifting exists was formulated by A. Haar for the Lebesgue measure on the real axis and was solved by J. von Neumann in 1931. It was D. Maharam who proved that every σ-finite measure admits lifting [270]. The general case is handled in [375]. For the results in 1.2.7 and 1.2.8 as well as other aspects of lifting theory see in [78, 239, 375]. 1.6.3. (1) In the history of functional analysis, the rise of the theory of ordered vector spaces is commonly attributed to the contribution of G. Birkhoff, L. V. Kantorovich, M. G. Kre˘ın, H. Nakano, F. Riesz, H. Freudenthal, et al. At present, the

42

Chapter 1

theory of ordered vector spaces and its applications constitute a noble branch of mathematics representing, in fact, one of the main sections of contemporary functional analysis. The theory with a vast field of applications is thoroughly covered in many monographs (see [15, 19, 23, 100, 139, 145, 162, 163, 170, 184, 185, 197, 230, 231, 242, 262, 282, 336, 341, 388, 399, 409, 411]). Observe also the surveys [60–62] each with a rich reference list. (2) The credit for finding the most important instance of ordered vector spaces, an order complete vector lattice or a K-space, is due to L. V. Kantorovich. This notion appeared in Kantorovich’s first fundamental article on this topic [153] where he wrote, “In this note, I define a new type of space that I call a semiordered linear space. The introduction of such a space allows us to study linear operations of one abstract class (those with values in such a space) as linear functionals.” Here L. V. Kantorovich stated an important methodological principle, the heuristic transfer principle for K-spaces. (3) The concept of universal completion for a K-space was introduced in another way by A. G. Pinsker (see [163]). He also proved existence of a universal completion unique to within isomorphism for an arbitrary K-space. The concepts of order-unity, fragment, and spectral function were introduced by H. Freudenthal. He also established Theorem 1.3.9 (1) (see [163, 388]). (4) Weak and strong Freudenthal properties for general vector lattices were introduced and studied B. Lavriˇc [234]. Using the principal ideal lattice he also introduced the notion of zero-dimensional vector lattice with strong unity and proved that a vector lattice has the weak Freudenthal property if and only if every principal ideal in it is zero-dimensional. M. Pannenberg [315] extended Lavriˇc’s notion of dimension zero to an arbitrary value of dimension by using the lattice Ip (E), see 1.3.5 (3). He also defined the topological stable rank tsr(E) for a Banach lattice E with quasi-interior positive elements and established that dim Ip (E) = tsr (E). (5) Sometimes the use of Freudenthal’s Spectral Theorem may be replaced by the following assertion: In a Banach lattice E with the principal projection property every order interval is the norm closed convex hull of its own extreme points. (The extreme points of the interval [0, e] are exactly the fragments of e.) The assertion remains valid if E is a Banach lattice with quasi-interior positive elements having the topological stable rank zero, see M. Pannenberg [315]. 1.6.4. (1) The fact that for a complete Boolean algebra B the set K(B) of resolutions of unity is a universally complete K-space with base isomorphic to B (see 1.4.3 and 1.4.4) is due to L. V. Kantorovich [163]. Theorem 1.4.4 was obtained by A. G. Pinsker (see [163]). The representation of an arbitrary K-space as an order-dense ideal in C∞ (Q) was established independently by B. Z. Vulikh and

Boolean Algebras and Vector Lattices

43

T. Ogasawara (see [163, 388]). (2) The theory of vector lattices and K-spaces arose historically before the theory of general ideal spaces which began developing in the 1950s with research by J. Diedonn`e, G. G. Lorentz, I. Galperin, H. W. Ellis, A. C. Zaanen, W. A. J. Luxemburg, et al. The synthesis of these theories occurred in the 1960s in the works of W. A. J. Luxemburg, A. C. Zaanen, and G. Ya. Lozanovski˘ı (see [61, 409]). After the papers of Diedonn`e, one of the most common terms for an ideal space is a K¨ othe space. The term “ideal space” was coined within the school of M. A. Krasnosel0 ski˘ı on integral operators and equations (P. P. Zabre˘ıko, P. E. Sobolevski˘ı, Ya. B. Rutitski˘ı et al.) due to the fact that an ideal space is actually an o-ideal in L0 (see [188, 189]). 1.6.5. (1) The material presented in 1.5.1–1.5.8 is traditional for the theory of normed lattices and can be found in [23, 162, 231, 242, 263, 336, 341, 409]. (2) Theorems 1.5.10, 1.5.11 and the equivalence (1) ⇔ (2) in 1.5.12 belong to A. I. Veksler and V. A. Ge˘ıler [381]. (2) In this book we never touch the theory of positive operators in a Banach space with cone which originated with M. G. Kre˘ın’s articles written in the 1940s and developed later by M. A. Krasnosel0 ski˘ı in Russia and by many research groups in other countries. The history, state of the art, and various applications are reflected in [35, 66, 139, 170, 184, 185, 187, 188, 285, 389, 390, 399].

Chapter 2 Lattice-Normed Spaces

In this chapter we consider structural properties of a vector space with some norm taking values in a vector lattice. Such a vector space is called a lattice-normed space; an LNS for short. The most important peculiarities of LNSs are connected with the decomposability property (2.1.1 (4)). The latter allows us, in particular, to indicate a complete Boolean algebra of linear projections in a lattice-normed space, which is isomorphic and closely related to the Boolean algebra of band projections of the norm lattice (2.1.3, 2.1.4). Moreover, a decomposable LNS admits a compatible module structure over a certain ring of orthomorphisms (2.1.8). These facts are closely related with the disjointness relation induced by the vector norm (2.1.2). If an LNS is simultaneously a vector lattice then there is another disjointness relation connected with its lattice structure. Some simple interrelation between them is reflected in the notions of norm-indecomposable and norm-n-decomposable elements (2.1.9). It turns out that every norm-n-decomposable elements is very often the sum of n norm-indecomposable elements (2.1.10). Partitions of unity in a Boolean algebra lead to the operation of mixing elements in a lattice-normed space (2.2.1). If there exists a mixing of every (vector) norm-bounded family in a lattice-normed space then the space is called disjointly complete (2.1.5). For instance, such are Banach–Kantorovich spaces (BKSs), i.e. decomposable and order complete latticenormed spaces (2.2.1). Moreover, a decomposable lattice-normed space is order complete if and only if it is disjointly (laterally) complete and complete with respect to relative uniform convergence (2.2.2). In addition, every LNS has a lattice norm completion that can be obtained by consecutively applying the operations of mixing and closure with respect to relative uniform convergence (2.2.8, 2.2.9). LNSs consisting of continuous and measurable vector-functions (Section 2.3) are most frequent in analysis. A more general example of an LNS is presented by a space of sections of a continuous Banach bundle (CBB) (2.4.7). It turns out that each BKS is linearly isometric to a space of almost global sections of some continuous Banach bundle (2.4.10). Thus, an LNS admits some functional representation.

Lattice-Normed Spaces

45

This representation is uniquely determined in the class of ample continuous Banach bundle (2.4.6). Every continuous Banach bundle over an extremal compact space has a unique ample hull to within isometry (2.4.11). The notion of ample continuous Banach bundle allows us, in particular, to introduce the CBB of operator spaces (2.4.13). The notion of measurable Banach bundle (MBB) is a measurable analog of a continuity structure. Measurable sections are defined as the limits of almost everywhere convergent sequences (on the subsets of finite measure) of elements of some set of sections given axiomatically and called a measurability structure (2.5.1). Under some assumptions the space of equivalence classes of measurable sections (2.5.2) of a measurable Banach bundle appears to be a bo-complete LNS (2.5.3). The notion of lifting in a factor space of measurable sections of an MBB is introduced (2.5.5). Measurable Banach bundles with lifting are called liftable. There is a constructive connection between liftable measurable Banach bundles and complete continuous Banach bundles. A method for constructing a liftable MBB given an arbitrary complete CBB over the corresponding extremal compact space is described in 2.5.8 and 2.5.9. Moreover, this method is universal: every liftable MBB is so obtainable (2.5.10). These facts enable us to represent Banach–Kantorovich spaces as the spaces of measurable functions associated with measurable Banach bundles (2.5.11–2.5.13). 2.1. Preliminaries In this section, we introduce basic notions and consider simple properties of vector norms. 2.1.1. Consider a vector space X and a real vector lattice E. (All vector lattices under consideration are assumed Archimedean.) A mapping · : X → E+ is a vector (E-valued ) norm if it satisfies the following axioms: (1) x = 0 ⇔ x = 0 (x ∈ X); (2) λx = |λ| x (λ ∈ R, x ∈ X); (3) x + y ≤ x + y (x, y ∈ X). A vector norm is called a decomposable or Kantorovich norm if (4) for all e1 , e2 ∈ E+ and x ∈ X, from x = e1 + e2 it follows that there exist x1 , x2 ∈ X such that x = x1 + x2 and xk = ek (k := 1, 2). In the case when condition (4) is valid only for disjoint e1 , e2 ∈ E+ , the norm is said to be disjointly-decomposable or, in short,

d-decomposable. A triple X, · , E in brief (X, E), X, · , or X with the default parameters
omitted is a lattice-normed space (over E) if · is an E-valued norm in the vector space X. The space E is called the norm lattice
of X. If the norm · is decomposable (d-decomposable) then the space X, · itself is called decomposable

46

Chapter 2

(d-decomposable). In the sequel we use the abbreviation LNS for “lattice-normed space.” 2.1.2. If x ∧ y = 0 then we call the elements x, y ∈ X disjoint and write x ⊥ y. As in the case of a vector lattice, a set of the form M ⊥ := {x ∈ X : (∀y ∈ M ) x ⊥ y}, with ∅ 6= M ⊂ X, is called a band or a component. The symbol B(X) denotes the set of all bands ordered by inclusion. We say that K ∈ B(X) is a projection band if K ⊕ K ⊥ = X. The projection h(π) onto the band K along the band K ⊥ is called a band projection. A lattice-normed space X is said to have the projection property whenever every band in X is a projection band. For uniformity, we often write B(X) instead of B(X) and take liberty of using the terminology from the theory of vector lattices. Such a treatment is admissible as long as X is considered as a LNS by itself. But, if X is simultaneously a vector lattice, we should be more accurate to avoid confusions, see  2.1.4 and 2.1.9 below. Given L ⊂ E and M ⊂ X, we let by definition h(L) := x ∈ X : x ∈ L and M := { x : x ∈ M }. It is clear that h(L) ⊂ L ∩ X . ⊥⊥

(1) Suppose that every band of the vector lattice E0 := X contains the norm of some nonzero element. Then B(X) is a complete Boolean algebra and ⊥⊥
and the mapping L 7→ h(L) is an isomorphism of the Boolean algebras B X B(X). C It is clear that the mapping h preserves the intersection of any nonempty family of bands. Therefore, h preserves infima, since in
the algebras under consider⊥⊥
ation they coincide with intersections. Moreover, h {0} = {0} and h X = X.
⊥⊥ Thus, it is sufficient to establish that h(L⊥ ) = h(L)⊥ for L ∈ B X . The inclusion h(L⊥ ) ⊂ h(L)⊥ is obvious. If 0 6= x ∈ h(L)⊥ then x is disjoint from all the elements of the form y in L. At the same time, x ∈ / h(L⊥ ) implies that ⊥⊥ 0 < e ≤ x for some e ∈ L+ . Therefore, in the band {e} there are no elements of the form y 6= 0, which contradicts our assumption. B (2) If elements x, y ∈ X are disjoint, then x + y = x + y . C Indeed, from the relations x ∧ y = 0 and x ≤ x + y + y we infer that x ≤

x+y + y




∧ x ≤ x+y ∧ x ≤ x+y .

Similarly, y ≤ x + y ; therefore, x + y = x ∨ y ≤ x+y . B (3) For every pair of disjoint elements e1 , e2 ∈ E the decomposition x = x1 + x2 with x1 = e1 and x2 = e2 is unique.

Lattice-Normed Spaces

47

C Assume x1 = y1 = e1 , x2 = y2 = e2 , and x = x1 + x2 = y1 + y2 . Then x1 − y1 ⊥ y2 − x2 , since x1 − y1 ≤ x1 + y1 = 2e1 and x2 − y2 ≤ 2e2 . By (2) 0 = (x1 − y1 ) + (x2 − y2 ) = x1 − y1 + x2 − y2 , whence x1 = y1 and x2 = y2 . B (4) Assume the hypotheses of Proposition (1) to be satisfied. Suppose that X is d-decomposable and there exists a band projection π onto L ∈ B(E0 ). Then the projection h(π) onto the band K := h(L) along the band K ⊥ exists and, moreover, π x = h(π)x for all x ∈ X. C In view of the d-decomposability condition, for every x ∈ X, there are x1 , x2 ∈ X such that x = x1 + x2 , x1 = π x , and x2 = π ⊥ x . This means that X is the direct sum of the bands K and K ⊥ . Let h(π) be the projection onto K along K ⊥ . By the definition of the isomorphism h, we have h(π)x ∈ K = h(πE0 ), i.e., h(π)x ∈ πE0 . Thus, π ⊥ h(π)x = 0, i.e., π h(π)x = h(π)x . Since h(π)x and h(π ⊥ )x are disjoint, in view of (2) we have
π x = π h(π)x + h(π ⊥ )x = π h(π)x . Consequently, π x = π h(π)x = h(π)x . B 2.1.3. In the sequel, by a Boolean algebra of projections in a vector space X we mean a set B of commuting idempotent linear operators that act in X. Moreover, the Boolean operations have the following form: π ∧ ρ := π ◦ ρ = ρ ◦ π,

π ∨ ρ = π + ρ − π ◦ ρ,

π ∗ = IX − π

(π, ρ ∈ B),

and the zero and identity operators in X serve as the zero and unity of the Boolean algebra B. ⊥⊥

Suppose that E0 := X is a vector lattice with the projection property and the space X is d-decomposable. Then X have the projection property. Moreover, there exists a complete Boolean algebra B of band projections in X and an isomorphism h from P(E0 ) onto B such that
b x = h(b)x b ∈ P(E0 ), x ∈ X . C A nonzero band L ∈ B(E0 ) cannot be disjoint from the set X . Hence, for some x ∈ X, we have x ∈ / L⊥ . If π is the projection onto L then the element π x differs from zero. Due to d-decomposability of X, we have x0 = π x ∈ L for some x0 ∈ X. Thus, we may apply 2.1.2 (1, 4). To every band K ∈ B(X) there corresponds the projection πK along K ⊥ . Assign B := {πK : K ∈ B(X)}. It is clear that B is a complete Boolean algebra of projections. With an order projection ρ ∈ P(E0 ), we associate the projection πK with K := h(ρE0 ). Denote the mapping thus obtained by the same letter h. Then h is an isomorphism of the Boolean algebras P(E0 ) and B. The required property of the isomorphism h follows from (3). B We identify the
Boolean algebras P(E0 ) and P(X) := B and write π x = πx x ∈ X, π ∈ P(E0 ) .

48

Chapter 2

2.1.4. Suppose that X is a vector lattice. The norm · is monotone if |x| ≤ |y| implies x ≤ y for x, y ∈ X. We say that · is order semicontinuous (osemicontinuous for short) if sup xα = sup xα for each increasing net (xα ) ⊂ X with a least upper bound x ∈ X and order continuous (o-continuous for short) if inf xα = 0 for any decreasing net (xα ) ⊂ X with inf α xα = 0. If X is a vector lattice with a monotone norm then P(X) consists of positive projections. If, in addition, X is order complete and the norm is order semicontinuous then P(X) is a complete subalgebra of the Boolean algebra P(X). C For a vector lattice with a monotone norm we have obviously that h(π) is an order ideal in X for any π ∈ P(E0 ). If 0 ≤ x ∈ h(π) and 0 ≤ y ∈ h(π ⊥ ) then x ∧ y ∈ π(E) ∩ π ⊥ (E) = {0} by monotonicity. Hence, x ∧ y = 0 and x and y are disjoint not only in the sense of Subsection 2.1.2 but also in the sense of the order relation in X. Therefore the projection operator corresponding to the decomposition X = h(π) ⊕ h(π ⊥ ) is positive. Now, assume that X is order complete and · is order semicontinuous. It suffices to prove that h(π) is order-closed in X. Let (xα ) ⊂ h(π)+ be a bounded increasing net. Since the net xα is bounded in E, by order semicontinuity of the norm we have sup xα = sup xα ∈ π(E). Therefore sup xα ∈ h(π) and h(π) is a band. B 2.1.5. We say that a net (xα )α∈A bo-converges to an element x ∈ X and write x = bo-lim xα if there exists a decreasing net (eγ )γ∈€ in E such that inf γ∈€ eγ = 0 and, for every γ ∈ € , there is an index α(γ) ∈ A such that x − xα ≤ eγ for all α ≥ α(γ). Given an element e ∈ E+ , let the following condition be satisfied: for every number ε > 0, there is an index α(ε) ∈ A such that x − xα ≤ εe for all α ≥ α(ε). Then we say that (xα ) br-converges to x and write x = br-lim xα . A net (xα ) is said to be bo-fundamental (br-fundamental ) if the net (xα − xβ )(α,β)∈A×A bo-converges (br-converges) to zero. A lattice-normed space is called bo-complete (br-complete) if every bo-fundamental (br-fundamental) net in it bo-converges (brconverges) to an element of the space. Take a family (xξ )ξ∈„ and associate with itP the net (yα )α∈A , where A := Pfin („) is the set of all finite subsets of „ and yα := ξ∈α xξ . If x := bo-lim yα exists then the family (x Pξ ) is said to be bo-summable and x is its sum. It is conventional to write x = bo- ξ∈„ xξ in this case. Sometimes, if X is not a vector lattice, we write o and r instead of bo and br in the terms like o-complete. A set M ⊂ X is called norm-bounded if there exists an e ∈ E+ such that x ≤ e for all x ∈ M . A space X is called disjointly complete or d-complete if every norm-bounded set in X of pairwise disjoint elements is bo-summable. 2.1.6. A subspace X0 ⊂ X is called a bo-ideal if, for x ∈ X and x0 ∈ X0 , from x ≤ x0 it follows that x ∈ X0 . In this subsection we assume that X is

Lattice-Normed Spaces

49

decomposable. (1) A subspace X0 ⊂ X is a bo-ideal if and only if X0 = h(L) for some o-ideal L ⊂ E. C Sufficiency is obvious. In order to prove necessity, we take an arbitrary oideal X0 ⊂ X. Let L be the bo-ideal generated by the set X0 . It is clear that X0 ⊂ h(L). If x ∈ h(L) then x ≤ u1 + · · · + un for suitable u1 , . . . , un ∈ X0 . Due to decomposability of X, we have the representation x = x1 + · · · + xn , where xk ≤ uk (k := 1, . . . , n). By the definition of bo-ideal, xk ∈ X0 ; hence, x ∈ X0 . Thus, X0 = h(L). B (2) The set h(L) minorizes X particular, 2.1.2 (1) holds.

⊥⊥

∩ L for every o-ideal L ⊂ E. In

⊥⊥

C Indeed, if 0 < e ∈ X ∩ L then there exists an element 0 < e0 ≤ e such that e0 ≤ x1 + · · · + xn for some x1 , . . . , xn ∈ X. Due to decomposability of X, the representation e0 = u1 + · · · + un holds, where u1 , . . . , un ∈ X. For at least one of the numbers k, we have uk 6= 0. Since e ∈ L and uk ≤ e, it follows that uk ∈ h(L). B (3) An order ideal K ⊂ X is a band if and only if it is bo-closed. C Necessity is obvious. Suppose that K is a closed bo-ideal. According to (1), ⊥⊥ we have K = h(L), where L is a bo-ideal in E. Demonstrate that K = h(L ).  ⊥⊥ If x ∈ h(L ) then x = sup E , where E := e ∈ L : 0 ≤ e ≤ x . By decomposability of X, we may choose an xe so that x − xe = x − e and xe = e. The set E is directed upward and the net (xe )e∈E o-converges to x. Consequently, x ∈ K and K ⊃ h(L⊥⊥ ). The reverse inclusion is obvious. B 2.1.7. Under the assumption that (X, E) is a decomposable LNS, we establish three auxiliary facts. (1) Suppose that an element x ∈ X and an increasing sequence (an ) ⊂ E+ are given and, moreover, (an ) ≤ x (n ∈ N). Then there exists a sequence (xn ) ⊂ X such that, for all n ∈ N and m > n, we have xn = an ,

x − xn = x − an ,

xm − xn = am − an .

C Denote bn := x − an (n ∈ N), b0 := x . By decomposability of X, we may choose sequences (un ) ⊂ X and (vn ) ⊂ X so that x = u1 + v 1 ,

u1 = a1 ,

v 1 = b1 ,

un+1 + vn+1 = vn , vn+1 = bn+1 , un+1 = bn − bn+1 . Pn Assign xn = k=1 uk . Then x = xn + vn and the following relations hold: xn ≤

n X k=1

uk =

n X k=1

(bk−1 − bk ) = b0 − bn = an .

50

Chapter 2

At the same time, x ≤ xn + vn ≤ an + bn = x ; consequently, xn = an . Thus, for m > n, we have xm − xn =

m X k=n+1

uk ≤

m X

uk = am − an ≤ xm − xn . B

k=n+1

(2) If (X, E) is d-complete then it has the projection property. In particular, every d-complete Archimedean vector lattice has the projection property. C Given y ∈ X, denote  by E(y) the o-ideal in E that is generated by the ele-
ment y . Assign X(y) := x ∈ X : x ∈ E(y) . It is easy to see that X(y), E(y) is a decomposable LNS. If K is a band in X and K(y) := K ∩ E(y) then projections of y onto K and K(y) exist simultaneously and they are equal. At the same time E(y) is disjointly complete. Indeed, for an arbitrary family (eξ ) that is bounded in E(y) and consists of disjoint elements, there is a family P P (xξ ) ⊂ X such that xξ = eξ . Therefore, for x = bo- xξ , we have x = o- xξ = sup eξ ∈ E(y). Thus, in view of 2.1.2 (3), it remains to apply 1.5.11. B ⊥⊥

(3) If (X, E) is d-complete and r-complete then E0 := X is a Kspace and X = E0+ . ^ be the Dedekind C According to (2) X has the projection property. Let E ^ +, completion of E. Since E(y) is d-complete, we deduce that, for every e ∈ E(y) there is an increasing sequence en ∈ E(y)+ that r-converges to e. Using (1), choose a sequence (xn ) ⊂ X such that xn = en and xm − xn ≤ em − en (m > n). It is clear that the sequence (xn ) is r-fundamental and, due to r-completeness of X, x = r-lim xn ∈ X exists. Moreover, x = r-lim en = e. Consequently, E(y) = ^ E(y). Next, let e be an arbitrary positive element in the Dedekind completion ⊥⊥ of the lattice X . Then, for every maximal disjoint family (eξ ) ⊂ X , we ^ so that π ξ e ≤ neξ , may choose a partition (πnξ ) of the identity projection in E n ξ ξ i.e., πn e ∈ X . Therefore, there is a family (xn ) ⊂ X and an x ∈ X such that ⊥⊥ xξn = πnξ e ≤ e and x = e. Consequently, E0 = X is a K-space and X = E0+ . B 2.1.8. Let A be a sublattice and subring in Orth(E) containing P(E). We say that a lattice-normed space X over E admits a compatible module structure over A if X can be endowed with a structure of a faithful unitary A-module such that: (a) the natural representation of A in X defines an isomorphism of the Boolean algebras P(E) and P(X) from 2.1.3; (b) ax = |a| x (a ∈ A, x ∈ X). In the case when X is a vector lattice we assume, in addition to (a) and (b), the following:

Lattice-Normed Spaces

51

(c) B(X) is a complete subalgebra of the Boolean algebra of bands B(X). Let X be a d-decomposable lattice-normed space over a vector lattice E with ⊥⊥ and let A be a sublattice and subring in Orth(E) containing P(E). E = X Each of the following assertions implies that X admits a compatible module structure over A: (1) A is the algebra of finite rank elements; (2) E is order σ-complete, A := Z (E), and X is br-complete; (3) E is order σ-complete, A = Orth(E), and X is sequentially bocomplete; (4) E is order complete, A = Orth(E), and X is a vector lattice with an order semicontinuous monotone norm. P C Let a ∈ A be a finite rank element, i.e. a = λk πk where λ1 , . . P . , λn ∈ R and π1 , . . . , πn is a finite partition of unity in P(E). Then we put ax := λk πk x. Taking into account 2.1.2 (2), 2.1.3, and the identification of the Boolean algebras P(E0 ) and P(X), ax =

X

λ k πk x =

X

|λk |πk x = a x .

Now an arbitrary a ∈ A is the order limit of an increasing sequence of finite rank elements (an ) ⊂ A. The sequence (an x) ⊂ X is bo-fundamental, since an x−am x = |an − am | x → 0. Therefore, we may assign ax := bo-lim an x. Moreover, ax = bo-lim an x = o-lim |an | x = a x . The remaining part of the proof is straightforward. B 2.1.9. Let X be a lattice-normed vector lattice over E. In this case we have two different disjointness relations in X, one induced by vector norm as defined in 2.1.2 and the other determined by the order relation in X, see 1.3.3. They will be referred to, if need be, as order disjointness and norm disjointness, respectively. Vn An elementPx ∈ X is said to be norm-n-decomposable if k=0 xk = 0 whenn ever |x| = k=0 xk and x0 , x1 , . . . , xn are pairwise order disjoint elements in X+ . A norm-1-decomposable element is called norm-indecomposable. If E is a vector lattice with the principal projection property then we may state the following equivalent definition: x is norm-n-decomposable if and only if for every collection of pairwise order disjoint elements x0 , x1 , . . . , xn ∈ X+ with x = x0 + x1 + · · · + xn there exists a partition of unity π0 . . . πn in P(F ) such that πk xk = 0 for all k = 0, 1, . . . , n. Indeed, e0 ∧ · · · ∧ en = 0 is equivalent to πk ek = 0 (∀k = 0, 1, . . . , n) for a suitable partition of unity π0 . . . πn in P(F ).

52

Chapter 2

In the case when E admits compatible module structure over Orth(E) we may give one more equivalent definition by replacing the partition of unity with the collection of positive orthomorphisms σ0 , σ1 , . . . , σn such that σ0 +σ1 +· · ·+σn = IF . (1) The sum of n norm-indecomposable elements is norm-n-decomposable element. C Let x = u1 + · · · + un where u1 , . . . , un are norm-indecomposable elements in X. Without loss of generality, we may assume that uk ≥ 0 (k := 1, . . . , n). Take pairwise order disjoint elements x0 , x1 , . . . , xn ∈ X with x0 + x1 + · · · + xn = u1 + · · · + un . Using the Riesz Decomposition Property 1.3.2 (13), choose uk,l ∈ X+ such that uk =

n X

uk,l (k := 1, . . . , n),

xl =

n X

uk,l (l := 0, 1, . . . , n).

k=0

l=1

From 1.3.2 (15) we deduce

0≤

n ^ l=0

≤

X

xl =

n X n ^ l=0 k=1

uk,l ≤

n X n ^

uk,l

l=0 k=1

uj(0),0 ∧ uj(1),1 ∧ · · · ∧ uj(n),n

j∈J

where J is the set of all functions j : {0, 1, . . . , m} → {1, . . . , m}. At least two indices in {j(0), . . . , j(n)} are equal, say j(r) = j(s) = m, 0 ≤ r, s ≤ n, r 6= s. Therefore, uj(0),0 ∧ uj(1),1 ∧ · · · ∧ uj(n),n ≤ um,r ∧ um,s = 0. The latter follows from the relations 0 ≤ um,r ≤ xr , 0 ≤ um,s ≤ xs , um = um,0 + · · · + um,n , since xr and xs are order disjoint and um is norm-indecomposable. B (2) Let E be a vector lattice with the principal projection property and X be a d-decomposable lattice-normed vector lattice over E. Let x ∈ X be a positive norm-n-decomposable element and x = x0 + x1 + · · · + xn−1 for some collection of pairwise disjoint x0 . . . xn−1 . If π is a band projection in E with  ⊥⊥ then y := π(x0 ) is norm-indecomposable and π(E) ⊂ x0 ∧ x1 ∧ · · · ∧ xn−1 z := π(x − x0 ) norm-(n − 1)-decomposable; moreover, y and z are disjoint fragments of x. C Let 0 ≤ y1 ≤ y, 0 ≤ y2 ≤ y, y1 ∧ y2 = 0, and y1 + y2 = y. Prove that y1 ∧ y2 = 0. Let πk be the band projection onto { xk }⊥⊥ and put π :=

Lattice-Normed Spaces

53

π0 π1 . . . πn−1 . Then n + 1 elements πx1 , . . . , πxn−1 , y1 , y2 are pairwise disjoint and their supremum is equal to πx1 + πx2 + · · · + πxn−1 + y1 + y2 = π(x − x0 ) + π(x0 ) = πx. Since any fragment of a norm-n-decomposable element is also norm-n-decomposable we obtain πx1 ∧
· · · ∧ πxn−1 ∧ y1 ∧ y2 = 0, whence πe = 0, e := π x1 ∧ · · · ∧ xn−1 ∧ y1 ∧ y2 . At the same time π ⊥ e = 0, since π ⊥ e ≤ π ⊥ y1 ≤ π ⊥ πx0 = 0, so that e = 0. It follows from this that π( y1 ∧ y2 ) = 0, because π is the band
⊥⊥ projection onto x0 ∧ · · · ∧ xn−1 . Thus, y1 ∧ y2 = π ⊥ ( y1 ∧ y2 ) ≤ π ⊥ πx0 = 0. Clearly y is a fragment of x. Suppose now z = z1 + · · · + zn and zk ⊥ xl (k 6= l). Then πx is the sum of n + 1 pairwise disjoint collection πx0 , z1 , , . . . , zn and πx0 ∧ z1 ∧· · ·∧ zn = 0, since πx is norm-n-decomposable. It follows from this that πe = 0, π( x0 ∧ z1 ∧· · ·∧ zn ) = 0. At the same time π ⊥ e = 0, so that e = 0. Consequently, π( z1 ∧ · · · ∧ zn ) = 0 and z1 ∧ · · · ∧ zn = 0. B 2.1.10. Theorem. Let E be a vector lattice with the principal projection property and X be a d-complete lattice-normed vector lattice over E. Then every norm-n-decomposable element x ∈ X there exist n norm-indecomposable elements x1 , . . . , xn such that x = x1 + · · · + xn . C The proof is by induction on n. The case n = 1 is a tautology. Suppose the claim is true for n − 1. By the Kuratowski–Zorn Lemma there is a maximal set P of pairwise disjoint band projections in E such that for each π ∈ P there Pn−1 exist n pairwise disjoint elements x0 , x1 , . . . , xn−1 ∈ X such that i=0 xi = x and π(E) ⊂ ( x0 ∧ x1 ∧ · · · ∧ xn−1 )⊥⊥ . Using 2.1.9 (2), we may construct a functions π 7→ xπ defined on P such that xπ is positive and norm-indecomposable, yπ := πx − xπ is positive norm-(n − 1)decomposable, xπ = πxπ , and xπ and yπ are disjoint fragments of W x. Since the family (xπ )π∈P is disjoint and X is d-complete we may define y := {xπ : π ∈ P} and z := x − y. Observe that y and z are disjoint fragments of x. By definition πy = xπ for every π ∈ P, so that, setting ρ := sup P, we obtain ρy = y and ρ⊥ y = 0. If y = y1 + y2 with y1 ∧ y2 = 0 then πy1 + πy2 = xπ for π ∈ P and π( y1 ∧ y2 ) = 0, since xπ is norm-indecomposable; thus ρ( y1 ∧ y2 ) = 0. But ρ⊥ ( y1 ∧ y2 ) ≤ ρ y = 0, whence y1 ∧ y2 = 0. Assume now that z = z0 +z1 +· · ·+ zn−1 for a disjoint collection z0 , z1 , . . . , zn−1 ∈ X+ and put e := z0 ∧ · · · ∧ zn−1 ). Then πe for every π ∈ P, because πz = yπ is n−1-decomposable. At the same time x = (z0 + y) + z1 + · · · + zn−1 and, taking into consideration the definition of P, we obtain π ⊥ ( z0 ∧ · · · ∧ zn−1 ) ≤ π ⊥ ( z0 + y ∧ z1 ∧ · · · ∧ zn−1 ) = 0. We conclude Vn−1 that i=0 zi = 0. Thus, we have proved that y is norm-indecomposable, z is normn − 1-decomposable and x = y + z and we are done in the case of positive x. For an

54

Chapter 2

arbitrary norm-n-indecomposable x the modulus |x| is also norm-n-indecomposable and by above proved fact |x| = y1 + · · · + yn with yk norm-indecomposable. In view of the Riesz Decomposition Property x+ = u1 + · · · + un , x− = v1 + · · · + vn , uk + vk = yk (uk , vk ∈ X; k := 1, . . . , n). The element xk := uk − vk is normindecomposable, since yk = |xk | and x = x1 + · · · + xn . B 2.2. Completion Here we briefly consider the questions of completing lattice-normed spaces. 2.2.1. Every decomposable o-complete lattice-normed space is called a Banach–Kantorovich space (a BKS for short). If a Banach–Kantorovich space is in addition a vector lattice and the norm is monotone then it is called a Banach– Kantorovich lattice. Let (X, E) be a Banach–Kantorovich space and, moreover, ⊥⊥ . According to 2.1.2 (2), the Boolean E= X
algebras P(E) and B(X) can be identified and π x = πx π ∈ P(E), x ∈ X . For every bounded family P (xξ )ξ∈„ in X and every partition of unity (πξ )ξ∈„ in P(X), the sum x := o- ξ∈„ πξ xξ exists. Moreover, x is a unique element in X satisfying the relations πξ x = πξ xξ (ξ ∈ „). C If e := sup xξ then, for α, β ∈ Pfin („), we have  X  X πξ xξ ≤ yα − yβ = πξ e ≤ e, ξ∈α4β

ξ∈α4β

P

where yγ := ξ∈γ πξ xξ and α4β is the symmetric difference of the sets α and β. Hence it is clear that the net (yα ) is bo-fundamental; hence, x := bo-lim yα exists. B This proposition, in particular, ensures d-completeness of all BKSs. Moreover, the definitions imply readily that every BKS is br-complete as well. Thus, an arbitrary BKS (X, E) is d-complete and br-complete; therefore, by 2.1.7 (3) we have ⊥⊥ and E+ = X . E= X 2.2.2. Let B be a complete Boolean algebra and let A be an arbitrary nonempty set. Assign n
B(A) := ν : A → B : (∀α, β ∈ A) α 6= β ⇒ ν(α) ∧ ν(β) = 0 o _ ∧ ν(α) = 1 . α∈A

Thus, B(A) is the set of all partitions of unity in B that are indexed by elements of the set A. If A is an ordered set then we may order the set B(A) as well:

ν ≤ µ ⇔ (∀α, β ∈ A) ν(α) ∧ µ(β) 6= 0 ⇒ α ≤ β ν, µ ∈ B(A) .

Lattice-Normed Spaces

55

It is easy to show that this relation is actually a partial order in B(A). If A is directed upward (downward) then so does B(A). Let Q be the Stone space of the algebra B. Identifying an element  ν(α) with a clopen subset of Q, we construct the mapping ν¯ : Qν → A, Qν := ∪ ν(α) : α ∈ A , by letting ν¯(q) = α whenever takes the value α on ν(α). Moreover, q ∈ ν(α). Thus, ν¯ is a step-function that
¯(q) . We shall use the set B(A) in the proof of ν ≤ µ ⇒ (∀q ∈ Qν ∩ Qµ ) ν¯(q) ≤ µ the following important completeness criterion. 2.2.3. Theorem. A decomposable lattice-normed space is order complete if and only if it is disjointly complete and complete with respect to relative uniform convergence. C Necessity was noted in 2.2.1. We will prove sufficiency. Suppose that a decomposable lattice-normed space X is d-complete and r-complete. According to ⊥⊥ 2.1.7 (3), we may assume without loss of generality that E := X is a K-space. Take an o-fundamental net P (xα )α∈A ⊂ X. Let B := P(X) be the base of X. Given ν ∈ B(A), assign xν := α∈A ν(α)xα . The element xν is well defined due to the d-completeness of X. A new net (xν )ν∈B(A) thus appeared. Show that the net is r-fundamental. Choose a decreasing net (eα )α∈A ⊂ E so that inf eα = 0 and xα − xβ ≤ eγ ≤ e whenever α, β ≥ γ (α, β, γ ∈ A), where e ∈ E. Take an arbitrary number ε > 0. There exist a partition of unity (ρξ )ξ∈„ in the algebra B and a mapping ϕ : „ → A such that ρξ eϕ(ξ) ≤ εe (ξ ∈ „). This is easily  deduced from the properties of o-convergence in a K-space. Assign πα := sup ρξ : ϕ(ξ) = α, ξ ∈ „ and πα = 0 in case α ∈ / im ϕ. Then (πα )α∈A is a partition of unity in B and πα eα ≤ εe (α ∈ A). Denote this partition of unity by ν(ε). Show that xν 0 − xν ≤ εe whenever ν, ν 0 ≥ ν(ε). Let ν = (ρα )α∈A and ν 0 = (τα )α∈A . If πα ρβ τγ 6= 0 then β, γ ≥ α; consequently, πα ρβ τγ xν − xν 0 = πα ρβ τγ xγ − ρβ τγ xβ = πα ρβ τγ xγ − xβ ≤ πα eα ≤ εe. Summing up over α, β, and γ, we find xν − xν 0 ≤ εe for ν, ν 0 ≥ ν(ε). Thus, the net (xν ) is r-fundamental and, due to the r-completeness, x := r-lim xν exists. It is clear that x − xν ≤ εe whenever ν ≥ ν(ε). For any fixed index γ ∈ A, construct a special partition of unity ν := (ρα )α∈A by letting ρα = πα in case α 6= γ and α 6= β, ργ = 0, and ρβ = πγ ∨ πβ , where β ∈ A and β ≥ γ. Then ν ≥ ν(ε) and πγ x − xβ = πγ ρβ x − xν = πγ x − xν ≤ εe.  Thus, πγ x − xβ ≤ εe for all β, γ ∈ A, β ≥ γ. Put cγ := sup x − xβ : β ≥ γ and c := inf{cγ : γ ∈ A}. Then πγ c ≤ πγ cγ ≤ εe. Since γ ∈ A and ε > 0 are arbitrary, we have c = 0; hence, o- lim x − xα = 0. B

56

Chapter 2

2.2.4. As before, let mE be a universal completion of a vector lattice E; moreover, let a unity 1 be fixed together with the respective multiplication structure ⊥⊥ = E. The operator-dual in mE. Take a lattice-normed space X with X ∗ ∗ space X is defined as follows. An operator x : X → mE belongs to X ∗ if and only if there exists an element 0 ≤ c ∈ mE such that hx, x∗ i := x∗ (x) ≤ c x

(x ∈ X).

The least element 0 ≤ c ∈ mE satisfying the indicated relation exists. This element is denoted by x∗ . It is easy to see that the mapping x∗ 7→ x∗ is an mE-valued norm in X ∗ and the following inequality holds: hx, x∗ i ≤ x x∗

(x ∈ X).

(1) If x∗ ∈ X ∗ then πx∗ = π x∗ for every projection π ∈ P(mE). C From the inequality hx, x∗ i ≤ c x it follows readily that hx, π
x∗ i ≤ πc x ; therefore, πx∗ ≤ π x∗ . On the other hand, hx, x∗ i ≤ πx∗ + π ⊥ x∗ x for all x ∈ X; hence, x∗ ≤ πx∗ + π ⊥ x∗ ≤ πx∗ + π ⊥ x∗ . Hence we deduce π x∗ ≤ πx∗ . B (2) The space X ∗ is d-complete and r-complete. C Take a partition of unity (πξ ) in P(mE), a family (x∗ξ ) ⊂ X ∗ ,P and an element ∗ x ∈ X. K-space there exist hx, x i := πξ hx, x∗ξ i and P In a∗ universally complete ∗ ∗ x i ≤ e x . Hence it is clear that x ∈ X ∗ . Since e := πξ xξ ; moreover, hx, P πξ x∗ = πξ x∗ξ , we have x∗ = πξ x∗ξ in view of (1). Thus, d-completeness is proven. Completeness with respect to relative uniform convergence is established below in a more general situation (see 4.2.1). B (3) The space X ∗ is a universally complete BKS. C Decomposability of X ∗ follows from (1) due to r-completeness, and it remains to refer to Theorem 2.2.3. B 2.2.5. According to 2.2.4 (3), the second operator-dual space, X ∗∗ , is a universally complete BKS. The canonical embedding κ : X → X ∗∗ is defined, as usual, by the formula hx∗ , κxi := κx(x∗ ) = hx, x∗ i (x∗ ∈ X ∗ ). The canonical embedding κ is a linear isometry. C We only need to show that κ preserves the norm, i.e., κx = x (x ∈ X). Observe first that κx ≤ x , since hx∗ , κxi ≤ x x∗ (x∗ ∈ X ∗ ). Next, since the operator · is sublinear, therefore, by the Hahn–Banach–Kantorovich Theorem, there exists an operator x∗ : X → mE such that x = hx, x∗ i and hy, x∗ i ≤ y (y ∈ X). Hence it is clear that x∗ ≤ 1; therefore, x = hx∗ , κxi ≤ κx x∗ ≤ κx . B

Lattice-Normed Spaces

57

2.2.6. By a universal completion (an order completion or, in short, a bocompletion) of a lattice-normed space
(X, E) we mean a universally complete BKS (Y, mE) respectively, a BKS (Y, oE) together with a linear isometry ı : X → Y such that each universally complete o-complete subspace of (Y, mE) respectively,
any decomposable o-complete subspace of (Y, oE) containing ıX coincides with Y . Here oE is a Dedekind completion of the vector lattice E, and mE is, as before, a universal completion of oE; moreover, we assume E ⊂ oE ⊂ mE, see 1.1.8. Given a set U in a lattice-normed space Y , we assign  r(U ) := y = br-lim yn : (yn )n∈N ⊂ U , n→∞  o(U ) := y = bo-lim yα : (yα )α∈A ⊂ U , n o X d(U ) := y = boπξ yξ : (yξ )ξ∈„ ⊂ U , ξ∈„

where A is an arbitrary directed set, (πξ ) is an arbitrary partition of unity in P(Y ), and the limits and the sums exist in Y . Let r0 (U ) be the part of r(U ) containing the limits of sequences in U converging with regulator 1, and let d0 (U ) be the part of d(U ) containing the finite sums. Observe the following simple relations: d0 (d0 (U )) = d0 (U ), d(d0 (U )) = d(U ). Now denote mX := rd(κX), where κ is the canonical embedding X → X ∗∗ of 2.2.5 and the operations d and r are calculated in the universally complete BKS (X ∗∗ , mE). 2.2.7. If Y is a decomposable BKS then, for any U ⊂ Y , the following hold: (1) dd(U ) = d(U ); (2) dr0 d(U ) = r0 d(U ); (3) rd(U ) = r0 d(U ); (4) rrd(U ) = rd(U ). C In each of the desired equalities, the inclusion ⊃ is obvious; therefore, the reasoning below concerns the reverse inclusion. (1): Take a family (yξ )ξ∈„ in d(U ) and a partitionPof unity (πξ )ξ∈„ in the algebra P(Y ). Represent an element yξ ∈ d(U ) as yξ = u∈U πu,ξ u, where (πu,ξ )u∈U P is a partition of unity in P(Y ) for each ξ ∈ „. Assign y := ξ∈„ πξ yξ and π(ξ,u) := πξ ◦ πu,ξ (ξ ∈ „, u ∈ U ). If (ξ, u) 6= (η, v) then π(ξ,u) ◦ π(η,v) = πξ ◦ πη ◦ πu,ξ ◦ πv,η = 0. At the same time, _ (ξ,u)∈„×U

π(ξ,u) =

_ ξ∈„

πξ ◦

_ n∈U

 πu,ξ = 1;

58

Chapter 2

consequently, (πλ )λ∈„×U is a partition of unity in P(Y ). Now observe that π(ξ,u) y = πu,ξ (πξ y) = πu,ξ (πξ yξ ) = πξ (πu,ξ u) = π(ξ,u) u. P Thus, y = (ξ,u) π(ξ,u) u and y ∈ d(U ). (2): Now let the family (yξ ) be included in r0 d(U ). Fix a number P ε > 0 and, for each ξ ∈ „, choose a uξ ∈ d(U ) so that yξ − uξ ≤ ε1. If u = πξ uξ and y is as above, then y − u ≤ ε1. However, due to (1), u ∈ d(U ); therefore, y ∈ r0 d(U ). (3): Suppose that a sequence (yn )n∈N in d(U ) converges to an element y ∈ Y with regulator e ∈ mE. Choose a partition of unity (πξ )ξ∈„ in P(mE) so that πξ e ≤ λξ 1 for suitable λξ > 0. Take an arbitrary ε > 0. For each P ξ ∈ „, there exists an index n(ξ) ∈ N such that πξ y − yn(ξ) ≤ ε1. Assign u = ξ∈„ πξ yn(ξ) and observe that u ∈ d(U ) due to (1). It remains to take account of the inequality y − u ≤ ε1, and we arrive at the conclusion y ∈ r0 d(U ). (4): First, note that the operation r0 is a topological closure and, therefore, r0 r0 (A) = r0 (A) for each A ⊂ Y . However, the latter can be easily proven directly. Next, applying (2) and (3), we may write rrd(U ) = rr0 d(U ) = rdr0 d(U ) = r0 dr0 d(U ) = r0 r0 d(U ) = r0 d(U ) = rd(U ). B 2.2.8. Theorem. Each lattice-normed space has a universal completion that is unique to within linear isometry. The space (mX, mE) serves as a universal completion for (X, E) and is referred to as the universal completion of (X, E) in the sequel. C The d-completeness of the space mX follows from 2.2.7 (2, 3), and its rcompleteness is clear from 2.2.7 (4). By Theorem 2.2.3, mX is a BKS. Moreover, mX is universally complete, since the operation d in the definition of mX is calculated in the universally complete space (X ∗∗ , mE), see 2.2.6. The embedding ı := κ : X → mX is a linear isometry (2.2.5). If Y is a decomposable BKS and ıX ⊂ Y ⊂ mX, then the values of the operations d and r are always in Y ; therefore, mX = rd(ıX) ⊂ rd(Y ) ⊂ Y ⊂ mX, i.e., Y = mX. Thus, mX is a universal completion for the space X. Assume that Y 0 is one more universal completion for X and let ı0 : X → Y 0 be the corresponding isometric embedding. The operator h := ı0 ◦ ı−1 : ıX → Y 0 is a linear isometry. We may extend h onto d(ıX) and next onto rd(ıX) preserving linearity and isometry. Moreover, h(mX) is a universally complete BKS and ı0 X ⊂ h(mX) ⊂ Y 0 ; consequently, h(mX) = Y 0 . B

Lattice-Normed Spaces

59

2.2.9. In this subsection X is a d-decomposable lattice-normed space over a vector lattice E possessing the principal projection property. (1) Let U ⊂ X and d0 (U ) = U . Then for every x ∈ X there exists
a net (uα )α∈A in U such that the net x − uα α∈A decreases and { x − uα : α ∈ A} = { x − u : u ∈ U }. In particular, o-limα∈A x − uα = inf u∈U x − u . C Take x ∈ X and introduce some equivalence relation and preorder in U by the following formulas: u∼v ⇔ x−u = x−v , u4v ⇔ x−u ≥ x−v . Since E is a lattice with the principal projection property we may choose a projection π ∈ P(X) such that π x − u + π ⊥ x − v = x − u ∧ x − v . The element w := πu + π ⊥ v lies in U since d0 (U ) = U . Moreover, x−w = x−u ∧ x−v , whence u 4 w and v 4 w. Thus, the preordered set (U, 4) is directed upward. Hence it is clear that the factor set A := U/∼, endowed with the factor order, is an upward-directed ordered set. Now, the desired net (uα )α∈A is defined by choosing uα ∈ α (α ∈ A). B We say that a set U ⊂ X approximates an element x ∈ X if inf u∈U x − u = 0 and that U approximates a set V ⊂ X if U approximates every element of V . In case if U approximates each x ∈ X we say that U is an approximating set. (2) If V approximates U and W approximates V , then W approximates U . C For an arbitrary u ∈ U , denote e := inf w∈W w − u and suppose that e 6= 0. Since inf v∈V v − u = 0, there exist v ∈ V and a band projection ρ such that ρ v − u < ρe/2. By the same arguments, inf w∈W w − v = 0 and there are w ∈ W and a band projection π with π w−v < πρe/2. Thus we arrive at the contradiction πρe 6 πρ w − u 6 πρ w − v + πρ v − u < πρe/2 + πρe/2 = πρe. B (3) Let U ⊂ and x ∈ X. Then U approximates x if and only if x is the bo-limit of some net of elements of d0 (U ).  C Indeed, if U approximates x then inf x − u : u ∈ d0 (U ) = 0. Thus, in view of (1) there is a net (uα )α∈A in d0 (U ) such that o-lim x − uα = 0. Conversely, if x is a bo-limit of a net containing in d0 (U ) then d0 (U ) approximates x. It remains to apply (2), since U approximates d0 (U ). B

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2.2.10. Assume that E is an order complete vector lattice, B is a complete Boolean algebra, and h : P(E) → B is a ring homomorphism. Say that a net (πα )α∈A in P(E) h-converges to zero and write h-limα∈A πα = 0 provided that o-limα∈A πα = 0 and o-limα∈A h(πα ) = 0 in the Boolean algebras P(E) and B respectively. We say that a net (eα )α∈A in E h-converges to e ∈ E and write h-limα∈A eα = e if the net (e  α )α∈A is bounded beginning with some member and h-limα∈A [d] (|eα − e| − d)+ = 0 for all positive d ∈ E. In this case, we call the element e the h-limit of the net (eα )α∈A . Finally, we say that a net (uα )α∈A in X h-converges to u ∈ X and write h-limα∈A uα = u if h-limα∈A uα −u = 0. In this case, we call the element u the h-limit of the net (uα )α∈A . For an arbitrary family  
+ (eξ )ξ∈„ in E+ , the notation h- inf ξ∈„ eξ = 0 means that inf ξ∈„ h [d] (eξ −d) =0 for all positive d ∈ E. (1) If h-limα∈A eα = e or h- inf ξ∈„ eξ = 0, then o-limα∈A eα = e or inf ξ∈„ eξ = 0, respectively. The converse is also true provided that the homomorphism h is o-continuous. Let U be a subset of an LNS X. We say that U h-approximates an element x ∈ X if h- inf u∈U x − u = 0. We say that U h-approximates a set V ⊂ X if U h-approximates every element of V . A subset of an LNS X is called happroximating if it h-approximates X. Every h-approximating set is approximating and, in case the homomorphism h is o-continuous, the notions of approximating and h-approximating set coincide. (2) Suppose that V h-approximates U and W h-approximates V . Then W h-approximates U . C Consider an arbitrary element u ∈ U , fix a
positive element d of the norm lattice, and assign b := inf w∈W h [d][( x − z − d)+ ] . By 2.2.9 (2), it is sufficient to establish the equality b = 0. For simplicity, we assume that h([d]) = 1. Suppose to the contrary that b 6= 0. Then, in view of inf v∈V h([( v − u − d/2)+ ]) = 0, there is an element v ∈ V such that b0 := b ∧ h([ v − u − d/2)+ ]) < b. Similarly, in view of the equality inf w∈W h([( w − v − d/2)+ ]) = 0, there is an element w ∈ W such that (b\b0 ) ∧ h([ w − x − d/2)+ ]) < (b\b0 ). It is easy to verify that w satisfies the inequality b ∧ h([( w − u − d)+ ]) < b, which contradicts the definition of b. B (3) A set U ⊂ X h-approximates an element x ∈ X if and only if x is the h-limit of some net in d0 (U ). C If U h-approximates x then, in
view of 2.2.9 (1), there exists a net (vα )α∈A in d0 (U ) such that the net x − vα α∈A decreases and { x − vα : α ∈ A} = { x − v : v ∈ d0 (U )}. It remains to observe that h-limα∈A x − vα = 0. Conversely, if x is the h-limit of a net in d0 (U ), then d0 (U ) h-approximates x. It remains to observe that U h-approximates d0 (U ) and to use (2). B

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2.2.11. (1) Theorem. For each lattice-normed space X, there exists a bocompletion which is unique to within linear isometry and is referred to as the bocompletion of X in the sequel. C Recall that E ⊂ oE ⊂ mE. Assign  ^ = x ∈ mX : x ∈ oE . X ^ is an order completion of the space X. B Then X We always assume that any lattice-normed space X is contained in its bo^ completion X. ^ of the space X, we have X ^ = rdX. If X (2) For the bo-completion X ⊥⊥ is decomposable and E0 := X is a vector lattice with the principal projection ^ property, then also X = oX. ^ ^ ∈ X C The first part of the assertion ensues from 2.2.7 and 2.2.8. Take x and choose a net (xα ) ⊂ X that bo-converges to x ^.
Apply Proposition 2.2.10 (1) ^ There exists a net xα in X such that the net with U := X and X := X. α∈A
x ^−xα α∈A decreases and e := inf x∈X x ^−x = inf α∈A x ^−xα , where the infimum ^ is calculated in oE. In view of the equality X = rdX, for an arbitrary number ε > 0,Pwe may find a family (xξ ) ⊂ X and a partition of unity (πξ ) ⊂ P(X) so that ^ − πξ xξ ≤ ε x . So, we may write x e = o-

X

X X ^ − xξ = x ^ − bo^. πξ e ≤ oπξ x πξ xξ ≤ ε x

^ = bo-lim xα . B Therefore, e = 0 and x 2.2.12. Let X be a vector lattice with a monotone norm. Then its norm ^ is a Banach–Kantorovich lattice and X is a sublattice in X. ^ completion X C Denote Y+ := rd(X+ ). Then Y+ is a cone. Indeed, Y+ is a wedge, since the operations r and d are linear. At the same time, if ±y ∈ Y+ then, by definition, for every 0 < ε ∈ R there are a partition of unity (πξ ) ⊂ P(E) and families (x0ξ ) and (x00ξ ) in X+ such that X y − boπξ x0ξ ≤ ε y ,

X y + boπξ x00ξ ≤ ε y .

From these inequalities we deduce X X 2y ≤ y − boπξ x0ξ + y + boπξ x00ξ ≤ 2ε y . Therefore, y = 0 and y = 0. Further details are obvious. B

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2.2.13. Let X be an arbitrary, not necessarily d-decomposable, lattice-normed space over an arbitrary vector lattice E. Suppose that a d-decomposable latticenormed space X over E includes X as a subspace with the induced norm. We say that X is a d-decomposable hull of X, if X = d0 (X), i.e., if every member of X is represented as a finite sum π1 x1 + · · · + πn xn , where x1 , . . . , xn ∈ X and pairwise disjoint π1 , . . . , πn ∈ P(E). (1) Suppose that a vector lattice E possesses the principal projection property. Then every (not necessarily d-decomposable) lattice-normed space over E has a d-decomposable hull that is unique to within an isometry. C Let mX be a universal completion of a lattice-normed space X. Define X Pn to be the set of all finite sums k=1 πk xk , where x1 , . . . , xn ∈ X and (πk )nk=1 is a partition of unity in the Boolean algebra P(E). It is easy to see that X is an LNS over E and is a d-decomposable hull of X. Uniqueness of this d-decomposable hull is obvious. B (2) We describe one more useful construction. Let E and F be K-spaces and let X be an LNS over
E. Suppose that a mapping S : E+ → F+ is subadditive
S(e1 + e2 ) 6 Se1 + Se2 , positively
homogeneous S(λe) = λSe (λ ∈ R+ ) , and increasing 0 6 e1 6 e2 ⇒ Se1 6 Se2 . Consider the vector subspace X0 := {x ∈ X : S( x ) = 0} and denote by SX x the coset in X/X0 containing an x ∈ X. It is easy to check that the space X/X0 is a lattice-normed space with respect to the norm SX x := S( x ). Observe that the LNS X/X0 need not be d-decomposable. We call a d-decomposable hull of the LNS X/X0 the norm transformation of X by means of S and denote it by SX. The linear operator SX : X → SX is called the operator of norm transformation of X by means of S. 2.3. Examples In this section we consider important examples of spaces of continuous, weakly continuous, measurable, and weakly measurable vector-functions that can be naturally normed by means of a vector lattice. 2.3.1. We begin with the simplest extreme cases, namely, vector lattices and normed spaces. If X = E then the modulus of an element can be taken as its vector norm: x := |x| = x ∨ (−x) (x ∈ E). Decomposability of this norm is easy from the Riesz Decomposition Property holding in every vector lattice. If E = R then X is a normed space. We shall use the conventional notation for the norm, k·k, and omit references to the order structure of the norm lattice. 2.3.2. Let Q be a topological space and let Y be a normed space. Let X := Cb (Q, Y ) be the space of bounded continuous vector-functions from Q into

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Y . Assign E := Cb (Q, R). Given f ∈ X, define its vector norm f by the relation f : t 7→ kf (t)k (t ∈ Q). Then · is a decomposable norm. Indeed, assume that f = h1 + h2 for some h1 , h2 ∈ E. Define a vector-function f1 : Q → Y so that f1 (t) = h1 (t)f (t)/kf (t)k whenever f (t) 6= 0 and f1 (t) = 0 whenever f (t) = 0. Then f1 ∈ X and f2 := f − f1 ∈ X; moreover, fk = hk (k := 1, 2). The space X is br-complete if and only if Y is a Banach space. 2.3.3. Suppose that E is an order-dense ideal of the universally complete vector lattice C∞ (Q), where Q is an extremal compact space. Let C∞ (Q, X) be the set of cosets of continuous vector-functions u that act from comeager subsets dom(u) ⊂ Q into some normed space X. Recall that a set is called comeager if its complement is meager. Vector-functions u and v are equivalent if u(t) = v(t) whenever t ∈ dom(u) ∩ dom(v). The set C∞ (Q, X) is endowed, in a natural way, with the structure of a module over C∞ (Q). Moreover, the continuous extension of the pointwise norm defines a decomposable norm on C∞ (Q, X) with values in C∞ (Q). Indeed, given any z ∈ C∞ (Q, X),
there exists a unique function xz ∈ C∞ (Q) such that ku(t)k = xz (t) t ∈ dom(u) , for every representative u of the coset z. Assign z := xz and  E(X) := z ∈ C∞ (Q, X) : z ∈ E . If X is a Banach space then E(X) presents a Banach–Kantorovich space with C∞ (Q, X) its universal completion. C The assertion can be easily deduced from 2.2.3. It also follows immediately from a more general fact which will be proved in the next section (see 2.4.6 and 2.4.7). B Denote by C# (Q, X) the part of C∞ (Q, Y ) consisting of cosets z with z ∈ C(Q). Thus, C# (Q, X) := E(X) where E := C(Q). Observe that C# (Q, X) is also a Banach–Kantorovich space, whereas the space C(Q, X) of everywhere defined continuous functions from Q to X, being a lattice-normed space over C(Q) (see 2.3.2) is not d-complete in general (see 2.4.9). Thus, C(Q, X) does not coincide with the space C# (Q, X), unless Q is finite or X is finite-dimensional. 2.3.4. For the same X and E define an embedding of the algebraic tensor Pn product E ⊗ X into the space E(X). Associate with an element z = k=1 ek ⊗ xk ∈ E ⊗ X the coset z¯ of the vector-function n n  n o X X q 7→ z¯(q) := ek (q)xk q ∈ Q(z) := s ∈ Q : |ek (s)| < +∞ . k=1

k=1

We also endow the space E ⊗ X with the norm n nX o z 0 := inf |ek |kxk k (z ∈ E ⊗ X), k=1

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Pn where the infimum is taken over all representations z = k=1 ek ⊗ xk . (1) The mapping ι : z 7→ z¯ from E ⊗ X into E(X) is a linear isometry. Pn C Clearly, ι is a linear embedding. If z = k=1 ek ⊗ xk then

n

n

X

X

z ≤ ek (·)xk ≤ |ek (·)|kxk k;

k=1

k=1

hence, z ≤ z 0 . For each P s ∈ Q(z) we may choose a function φs ∈ E such n 0 that φs (s) = 1. Put x := s k=1 ek (s)xk ∈ X and ek := ek − φs ek (s). Then Pn z = φs xs + k=1 e0k xk and for an arbitrary s ∈ Q(z) we have n   X z 0 (s) ≤ |φs |kxs k + |e0k |kxk k (s) k=1



= |φs (·)|kxs k +

n X

 |ek (·) − φs (·)ek (s)|kxk k (s)

k=1

X

n

= kz(s)k = z . B e (s)x = |φs (s)|kxs k = k k

k=1

Thus, E ⊗ X can be identified with a subspace of the LNS E(X). It turns out that E ⊗ X is, in a sense, dense in E(X).
(2) For every z ∈ E(X) and every ε > 0, there exist a family Q(ξ) ξ∈„ of pairwise disjoint clopen sets Q(ξ) ⊂ Q whose union is dense in Q and a family (zξ )ξ∈„ in E ⊗ X such that z|Q(ξ) − zξ |Q(ξ) ≤ ε z (ξ ∈ „).  ⊥⊥ C It is sufficient to consider the case in which z = E and the identical unity is contained in E. For every point q ∈ dom(z) ⊂ Q, choose a clopen neighborhood U (q) so that
kz(q) − z(q 0 )k ≤ εkz(q 0 )k q 0 ∈ U (q) . In the
Boolean algebra of clopen subsets of Q there exists a partition of unity Q(ξ) ξ∈„ which possesses the following property: for each ξ ∈ „, there is a point qξ ∈ Q such that Q(ξ) ⊂ U (qξ ). Now, if we assign zξ := χQ(ξ) z(qξ ) then
kz(q 0 ) − zξ (q 0 )k = kz(q 0 ) − z(qξ )k ≤ εkz(q 0 )k q 0 ∈ Q(ξ) . Thus, χQ(ξ) z − zξ ≤ ε z , which is equivalent to the desired inequality. B

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2.3.5. Now we turn to defining a space of weakly continuous vector-functions which is similar to E(X). We suppose that Z ⊂ X 0 is a norming subspace (for X), i.e., kxkX = sup{|hx, zi| : z ∈ Z, kzk ≤ 1} (x ∈ X). Here, as usual, X 0 is the dual space and h· , ·i is the canonical bilinear form of the duality X ↔ X 0 . Denote by M the set of σ(X, Z)-continuous vector-functions u : dom(u) → X such that dom(u) is a comeager set in Q. Consider the factor set C∞ (Q,
X|Z) := M /∼, where u ∼ v means that u(t) = v(t) t ∈ dom(u) ∩ dom(v) . The set C∞ (Q, X|Z) can be made into a vector space in the natural way: if u e stands for the cosets of u ∈ M then a sum λe u + µe v is interpreted as the coset of the pointwise sum λu(t) + µv(t), t ∈ dom(u) ∩ dom(v). For u ∈ M and z ∈ Z denote by hu, zi the unique continuous extension of the function t 7→ hu(t), zi (t ∈ dom(u)) onto the entire Q. If u ∼ v then clearly hu, zi = hv, zi; thus, for w ∈ C∞ (Q, X|Z) and an arbitrary u ∈ w we may put hw, zi := hu, zi. The set R(u) := hu, zi : z ∈ Z, kzk ≤ 1 is order-bounded in C∞ (Q), it is pointwise bounded on the comeager set dom(u). Therefore, for an arbitrary u ∈ w we may assign  w := u := sup hu, zi : z ∈ Z, kzk ≤ 1 , where supremum is taken in C∞ (Q). Observe that the function ku(·)k : t 7→ ku(t)k (t ∈ dom(u)) is pointwise the least upper bound of the same family R(u). Therefore, the functions u and ku(·)k coincide on a comeager subset of Q. Nevertheless, these function may differ on dom(u). It is easy to see that · is a decomposable norm with values in C∞ (Q). Moreover, C∞ (Q, X|Z) is naturally endowed with the structure of a faithful module over the ring C∞ (Q). Put  Ew (X, Z) := u ∈ C∞ (Q, X|Z) : u ∈ E . Indicate an important particular case Ew (X 0 ) := Ew (X 0 , X) that appears when X := X 0 and Z := X ⊂ X 00 . If X is a Banach space then for every order-dense ideal E ⊂ C∞ (Q) the set Ew (X, Z), endowed with the operations and E-valued norm · induced from C∞ (Q, X|Z), is a Banach–Kantorovich space over E and C∞ (Q, X|Z) is its universal completion. In particular, Ew (X 0 ) is a Banach–Kantorovich space over E. 2.3.6. Let X, Q, and E be the same as in 2.3.5. Let Y be one more normed space, with Z ⊂ Y 0 a norming space for Y and L (X, Y ) standing for the space of all bounded linear operators from X to Y . Denote by MQ (X, Y ) the set of all operator-functions K : dom(K) → L (X, Y ) that satisfy the following conditions:

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(a) Q0 := dom(K) is a comeager subset of Q; (b) the function s 7→ hK(s)x, zi (s ∈ Q0 ) is continuous for all x ∈ X and z ∈ Z. It follows from the definition that, given x ∈ X and z ∈ Z, there exists a unique element u ∈ C∞ (Q) for which u(t) = hK(t)x, zi (t ∈ Q0 ). Assign hKx, zi := u. Just as in 2.3.5 there exists a ϕ ∈ C∞ (Q) such that kK(s)k ≤ ϕ(s) (s ∈ Q0 ). Thus, it is clear that  K := sup hKx, zi : kxk ≤ 1, kzk ≤ 1 ≤ ϕ exists in the C∞ (Q). Introduce an equivalence in MQ (X, Y ) by letting K ∼ L e is the coset of an operatorwhenever K(s) = L(s) for all s ∈ dom(K)∩dom(L). If K e function K then we assign K := K . This definition is sound since K = L for equivalent K and L. Now introduce the space
 Ew L (X, Y ), Z := K ∈ MQ (X, Y )/∼ : K ∈ E .
It is easy to verify that Ew L (X, Y ), Z is a decomposable LNS. It coincides with Ew (X, Z) whenever X := L (X, Y ) and Z := X ⊗ Z. An important particular case is presented by the space


^ Y )0 , Ew L (X, Y 0 ) := Ew L (X, Y 0 ), Y := Ew (X ⊗ ^ Y is the projective tensor product of the spaces X and Y . where X ⊗ If Y is a Banach space then for every order-dense ideal E ⊂ C∞ (Q) the set
Ew L (X, Y ), Z is a Banach–Kantorovich space over E
and C∞ (Q, L (X, Y )|Z) 0 is its universal completion. In particular, Ew L (X, Y ) is a Banach–Kantorovich space over E. 2.3.7. Let (Š, †, µ) be a measure space with the direct sum property, let E be an order-dense ideal in L0 (Š, †, µ), and let X be a normed space. Let L0 (µ, X) := L0 (Š, †, µ, X) be the space of cosets of Bochner µ-measurable vectorfunctions acting from Š into X. As usual, vector-functions are equivalent if they assume equal values at almost all points of the set Š. If u e ∈ L0 (µ, X) is the coset of a measurable vector-function of u : Š → X then t 7→ ku(t)k (t ∈ Š) is a scalar measurable function whose coset is denoted by the symbol ze ∈ L0 (µ). Assign by definition E(X) := {u ∈ L0 (µ, X) : u ∈ E}.
Then E(X), E is a lattice-normed space with decomposable norm. Obviously, Lp (µ, X) (1 ≤ p ≤ ∞) coincides with E(X), where E = Lp (µ). If X is a Banach space then E(X) is a Banach–Kantorovich space and L0 (µ, X) is its universal completion. C The assertion can be easily deduced from 2.2.2. It also follows immediately from a more general fact which will be proved in Section 2.5 (see 2.5.4 (3)). B

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2.3.8. Suppose, as above, that (Š, †, µ) is a measure space with the direct sum property. Let ρ be a lifting of L∞ (Š, †, µ) and τ : Š → Q be the corresponding canonical immersion of Š into the Stone space Q of the Boolean algebra B(Š, †, µ), ∗ see 1.4.9. Denote by τ ∗ := τX the mapping that sends each function f ∈ C∞ (Q, X) to the equivalence class of the measurable vector-function f ◦ τ . (1) The mapping τ ∗ is a linear isometry (in the sense of vector norms) from C∞ (Q, X) onto L0 (Š, †, µ, X). The image of C# (Q, X) under τ ∗ coincides with the subspace L∞ (Š, †, µ, X). C A more general fact is proved in 2.5.9. B The algebraic tensor product E ⊗X is identified with the set of vector-functions of the form n X f (t) = ei (t)xi (ei ∈ E, xi ∈ X). i=1

It is clear that E ⊗ X is a linear subset in E(X). (2) Let E be an ideal space and let X be a Banach space. Then E ⊗ X is dense in E(X) with respect to the vector norm. C It is readily apparent from (1) and 2.3.4. B 2.3.9. Now we consider the lattice-normed space E[F ] of measurable functions of two variables defined on the product of some measure spaces (Š, †, µ) and (Š0 , †0 , µ0 ). To avoid technical inconveniences assume that (Š0 , †0 , µ0 ) is σ-finite; however, the definition and main properties remain valid in the general case. Let E be an ideal space on (Š, †, µ) and let F be a Banach ideal space on 0 (Š , †0 , µ0 ). Denote by E[F ] the space of all measurable functions K on Š0 × Š satisfying the following: (a) the coset of the function s 7→ K(s, t) belongs to F for almost all t ∈ Š; (b) the function t 7→ kK(·, t)kF is measurable and its coset K belongs to E. (1) Suppose that a norm on F is o-semicontinuous. Then E[F ] is a Banach–Kantorovich space over E and an ideal space on Š0 × Š. C There are two subtle reasonings in the proof. The first is to demonstrate that o-semicontinuity of the norm in F provides measurability for the function K (see [162, XI.1.4]). It follows from this that E[F ] is a linear space and an ideal space on Š0 × Š. Thus, E[F ] is also a decomposable d-complete lattice-normed space over E. The second is to prove bo-completeness of E[F ]. In virtue of Theorem 2.2.3 we need only to check that this space is br-complete. The latter is equivalent to br-completeness of all spaces Ee [F ] where e ∈ E+ and Ee := E(e). Define a scalar norm ||| · ||| on Ee [F ] by |||K||| := k K ke . In 7.1.2 we will prove that Ee [F ] is brcomplete if and only if (Ee [F ], ||| · |||) is norm complete. It can be also easily seen that Ee is an AM -space and (Ee [F ], ||| · |||) is a normed lattice. In view of 1.5.1 (5),

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it remains to prove that for every disjoint sequence P∞ (Kn ) of positive functions in Ee [F ] there exists K := supn Kn provided that n=1 |||Kn ||| < ∞. The sought least upper bound K exists in a universally complete vector space L0 (Š0 × Š, ν × µ). P∞ Show that K ∈ Ee [F ]. Observe that the series n=1 Kn is absolutely summable and its sum exists, since (Ee , kP· ke ) is norm complete. From this we deduce that ∞ for almost all t ∈ Š the series n=1 Kn (·, t) is absolutely summable in F . If Kt (·) stands for the sum of this series then Kt (s) = K(s, t) for almost allP s ∈ Š0 in view ∞ of 1.5.8. Thus K(·, t) ∈ F for almost all t ∈ Š. Moreover, K ≤ n=1 Kn and K ∈ Ee . B (2) Let F be a Banach ideal space on (Š0 , †0 , µ0 ) with an order semicontinuous norm. For every measurable vector-function f : Š → F , there is a measurable function K(s, t) on (Š0 × Š) such that, for almost all t ∈ Š the equality K(s, t) = (f (t))(s) holds for almost all s ∈ Š0 . The correspondence f 7→ K implements an isometric embedding of E(F ) onto a closed subspace and a sublattice of E[F ]. C Without loss of generality we may assume that (Š, †, µ) is a finite measure space. Approximate f by a sequence (fn ) of finite-valued functions in the sense of almost everywhere norm convergence in F : kfn (t) − f (t)kF → 0 for almost all t ∈ Š. Every function fn generates a measurable function Kn (·, ·) by formula Kn (s, t) = (fn (t))(s); moreover, Kn − Km (t) = kfn (t) − fm (t)kF → 0 almost everywhere. en − K e → 0 where L e ∈ E[F ] with K e Since E[F ] is norm o-complete there is K stands for the coset of measurable function L. Now the required identity follows from the relations: kf (t) − K(·, t)kF ≤ kf (t) − fn (t)kF + Kn − K (t) → 0. B (3) Let the measure µ be not purely atomic. Then the lattice-normed spaces E(F ) and E[F ] coincide (under embedding (2)) if and only if F is a Banach ideal space with order continuous norm. 2.3.10. Now we introduce the measurable version of the space Ew (X). Take the same E and X as in 2.3.7, and a norming subspace Z ⊂ X 0 , see 2.3.5. A vectorfunction u : Š → X is called σ(X, Z)-measurable or simply Z-measurable if, for each z ∈ Z, the function t 7→ hu(t), zi (t ∈ Š) is measurable. Denote the coset of the last function by the symbol hu, zi, so that hu, zi ∈ L0 (µ). Let M (Š, X|Z) be the set of Z-measurable vector-functions u : Š → X. We say that Z-measurable vector-functions u, v are Z-equivalent and write u ' v if measurable functions hu, zi and hv, zi are equal almost everywhere for each z ∈ Z. Consider the factor set

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L0 (µ, X|Z) := L0 (Š, †, µ, X|Z) := M (Š, X|Z)/ ' and define vector space structure in it by setting αe u + βe v := (αu + βv)∼ . For a coset u e ∈ L0 (µ, X|Z) with u ∈ M (Š, X|Z) put he u, zi := hu, zi.  Observe that the set R(e u) := he u, zi : z ∈ Z, kzk ≤ 1 is order-bounded in L0 (µ). Otherwise, according to 1.4.11 (2), one can choose a sequence (fn ) in R(e u) which is unbounded from above. But this is a contradiction, since the function f (t) := supn fn (t) is measurable and |f (t)| ≤ ku(t)kX < ∞ (t ∈ Š). Given u e∈ M / ', assign  u := sup hu, zi : z ∈ Z, kzk ≤ 1 , where the supremum is taken in the L0 (Š, †, µ). It is easy to verify that L0 (µ, X|Z) is a decomposable lattice-normed space over L0 (µ). Define now the set  Ew (X, Z) := u ∈ L0 (µ, X|Z) : u ∈ E . In the same manner as in 2.3.7, we indicate an important particular case, when X is a dual Banach space (X := X 0 ) and Z is its predual (Z := X ⊂ X 00 ). The notation Ew (X 0 ) := Ew (X 0 , X) is conventional. (1) For every order ideal E ⊂ L0 (µ) the space Ew (X 0 ) endowed with the operations and E-valued norm · induced from L0 (Š, †, µ, X 0 |X) is a Banach– Kantorovich space over L0 (Š, †, µ) and L0 (Š, †, µ, X 0 |X) is its universal completion. It is evident that every measurable vector-function is weakly measurable and u ∼ v implies u ' v for every pair of measurable vector-functions u and v. Therefore, there is a mapping (called the canonical embedding) assigning to each u e ∈ L0 (µ, X) the coset {v ∈ M (µ, X|Z) : v ' u} ∈ L0 (µ, X|Z). (2) The canonical embedding L0 (µ, X) → L0 (µ, X|Z) is a linear isometry. 2.3.11. Take the same E and X as in 2.3.10. Let Y be one more normed space and Z ⊂ Y 0 be a norming subspace. An operator-function K : Š → L (X, Y ) is called Z-weakly measurable if, for all x ∈ X and z ∈ Z, the function hz, Kxi : t 7→ hz, K(t)xi (t ∈ Š) is measurable. Denote by Mµ (X, Y ) the set of all Z-weakly measurable operator-functions K : Š → L (X, Y ). Introduce some equivalence relation ∼ in Mµ (X, Y ) by letting K ∼ L if and only if, for each pair (x, z) ∈ X × Z, the measurable functions hz, Kxi and hz, Lxi coincide almost everywhere. For a K ∈ Mµ (X, Y ), as well as for the corresponding e assign coset K,  e := sup hz, Kx i : kxk ≤ 1, kzk ≤ 1 , K = K

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where the supremum is taken in L0 (µ). This definition is sound, since the same reasoning as in 2.3.10 shows that K is the least upper bound of an order-bounded set in L0 (µ). Now put
 Ew L (X, Y ), Z := K ∈ Mµ (X, Y )/∼ : K ∈ E .
This notation agrees with 2.3.10, since it is easily seen that Ew L (X, Y ), Z = Ew (X, Z) whenever X := L (X, Y ) and Z := X ⊗ Z. We also note that Ew (X 0 ) =
Ew L (X, R) . Again, we distinguish an important particular case


^ Y )0 , Ew L (X, Y 0 ) := Ew L (X, Y 0 ), Y := Ew (X ⊗ ^ Y is the projective tensor product of the spaces X and Y . where X ⊗ The following assertions hold:
(1) Ew L (X, Y ), Z is a decomposable lattice-normed space;
(2) Ew L (X, Y 0 ) is a Banach–Kantorovich space. 2.3.12. An operator-function K : Š → L (X, Y ) is called simply measurable if, for each x ∈ X, the vector-function Kx : t 7→ K(t)x (t ∈ Š) is measurable. It is clear that any simply measurable operator-function is weakly measurable. Let Msµ (X, Y ) be the part of Mµ (X, Y ) consisting of simply measurable operator-functions with
values in L (X, Y ). As usual, we assume that Y ⊂ Z 0 and L (X, Y ) ⊂ L (X, Z 0 ). Assign
 Ews L (X, Y ) := K ∈ Msµ (X, Y )/≈ : K ∈ E , where K is the same as in 2.3.11 and the equivalence ≈ is defined as follows: L ≈ K if and only if, for each x ∈ X, the measurable vector-functions Kx and Lx coincide almost everywhere. If K ∈ Msµ (X, Y ) then we also have  K = sup kKxk : kxk ≤ 1 , where kKxk is
the measurable function t 7→ kK(t)xk (t ∈ Š). Observe that E(Y ) =
Ews L (R, Y ) and Ew (X 0 ) = Ews L (X, R) . 2.4. Continuous Banach Bundles The main subject of this section is the following assertion: Every Banach– Kantorovich space is linearly isomorphic to the space of almost global sections of some continuous Banach bundle.

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2.4.1. First of all, we give a general definition of a Banach bundle. Let Q be a topological space. A bundle over Q is defined to be a continuous surjection σ : X → Q of a topological space X onto Q. A nonempty set Xq = σ −1 (q) is called the stalk at a point q ∈ Q. A mapping s from a nonempty set dom(s) ⊂ Q into X is called a section over dom(s) if s(q) ∈ Xq for each q ∈ dom(s). A continuous section s is called local, almost global, or global, whenever its domain of definition dom(s) is respectively an open proper subset, a comeager subset, or the whole of Q. A bundle σ : X → Q is often identified with the mapping q 7→ Xq (q ∈ Q), and the notation X (q) is used instead of Xq . Moreover, the bundle is sometimes denoted by X , with the default parameters σ and Q omitted. A set of sections S is called stalkwise dense in X if the set {s(q) : s ∈ S} is dense in X (q) for every q ∈ Q. The product of two bundles σ : X → Q and σ 0 : X 0 → Q is the bundle τ : X ×Q X 0 → Q defined by the following formulas:  X ×Q X 0 := (x, x0 ) ∈ X × X 0 : σ(x) = σ 0 (x0 ) ,
τ : (x, x0 ) 7→ σ(x) = σ 0 (x0 ) (x, x0 ) ∈ X ×Q X 0 . 2.4.2. A continuous Banach bundle over Q is defined to be a bundle σ : X → Q that satisfies the following four conditions: (1) every stalk X (q) is a Banach space with the norm k·kq , and the topology defined by the norm coincides with that induced; (2) addition, X ×Q X 3 (x, y) 7→ x + y ∈ X , scalar multiplication, R × X 3 (α, x) 7→ αx ∈ X , and the zero section, Q 3 q 7→ 0q ∈ X (q), are continuous mappings; (3) the sets of the shape  U (s, ε) := x ∈ X : σ(x) ∈ dom(s), kx − s(σ(x))kσ(x) < ε , where ε > 0 and s is a local section, form a base for the topology on X ; (4) for each point x ∈ X and every number ε > 0, there exists a local section s such that σ(x) ∈ dom(s) and kx − s(σ(x))kσ(x) < ε. In the case of a paracompact space Q, the above definition can be simplified. Namely, instead of (3) and (4) we may require the following: (30 ) every neighborhood of zero 0q ∈ Xq contains a neighborhood of the form U (s, ε), where ε > 0 and s is the restriction of the zero section q 7→ 0q to some open set; (40 ) the mapping σ is open, and k·k : X → R, kxk := kxkσ(x) , is upper semicontinuous. In this case a continuous Banach bundle has a very useful property: (5)
For each x ∈ X , there exists a global continuous section s such that s σ(x) = x.

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2.4.3. It is sometimes convenient to define a topology for a Banach bundle with the help of a certain set of sections. Take a surjective mapping σ : X → Q and assume that Xq := σ −1 (q) is a Banach space for each q ∈ Q. The latter amounts to considering a mapping q 7→ Xq from Q into the class of Banach spaces. Introduce the structure of a vector space in the set of all global sections S(Q, X ) by letting (αu + βv)(q) = αu(q) + βv(q) (q ∈ Q) for α, β ∈ R and u, v ∈ S(Q, X ). For each section s ∈ S(Q, X ) we consider its pointwise norm |||s||| : q 7→ ks(q)kq (q ∈ Q). Now let C be a set of global sections that satisfies the following conditions: (1) C is a vector subspace of S(Q, X ); (2) for each s ∈ C , the function |||s||| is continuous;  (3) for each q ∈ Q, the set s(q) : s ∈ C is dense in the stalk Xq . With the help of C , called a continuity structure in X , we may introduce  a topology in X by taking the family U (s, ε) as a basis for the topology, where ε > 0 and s is the restriction of some section in C to an arbitrary open set. Then σ : X → Q is a continuous Banach bundle. Moreover, a section u ∈ S(Q, X ) is continuous if and only if the function q 7→ ku − skq (q ∈ Q) is continuous for every section s ∈ C . Let C(Q, X ) denote the space of all global continuous sections of X . 2.4.4. Consider an important example of a Banach bundle. Assume that
X, (k·kq )q∈Q is a multinormed space, where  Q is still a topological space. For each q ∈ Q, assign Xq := x ∈ X : kxkq = 0 and denote by Xq the completion of the normed factor space X/Xq . With each element x ∈ X associate a global section ı(x) ∈ S(Q, X ) by the formula ı(x)(q) = ϕq (x), where ϕq : Xq → X/Xq ⊂ Xq is the factor homomorphism. The set C := ı(X) satisfies conditions 2.4.3 (1–3) if and only if, for each x ∈ X, the function q 7→ kxkq (q ∈ Q) is continuous. If the last condition is satisfied then, due to 2.4.3, X becomes a continuous Banach bundle. In this case, the constructed Banach bundle X is said to be associated
with the multinormed space X, (k·kq )q∈Q , and the mapping ı : X → C(Q, X ) is called the canonical embedding. 2.4.5. We shall need the following auxiliary fact: Let (Qξ )ξ∈„ be a family of pairwise disjoint open sets whose union S is dense in Q. If Pξ is a comeager subset of Qξ for each ξ ∈ „, then the union P := ξ∈„ Pξ is again a comeager set in Q. C ForTeach ξ ∈ „, choose a sequence (Gn,ξ )n∈N of open dense subsets of Qξ such that n∈N Gn,ξ ⊂ Pξ (ξ ∈ „). Since Qξ are pairwise disjoint, we have [ \ \ [ Gn,ξ = Gn,ξ . ξ∈„ n∈N

n∈N ξ∈„

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S T Hence it is clear that if Gn := ξ∈„ Gn,ξ then P ⊃ n∈N Gn . Thus, P is a comeager set, since Gn is open and dense in Q for each n ∈ N. B 2.4.6. Now suppose that Q is an extremal compact space. Take a continuous Banach bundle X over Q. If u is an almost global section of the bundle X then the function q 7→ ku(q)kq is defined and continuous on a comeager set dom(u). Consequently, there
exists a unique function u ∈ C∞ (Q) such that u (q) = ku(q)kq q ∈ dom(u) . Introduce an equivalence relation in the set of almost global sections M(Q, X ) by letting u ∼ v if u(q) = v(q) whenever q ∈ dom(u) ∩ dom(v). ~ := u , where For equivalent u and v, we have u = v ; therefore, we may define u u ~ is the coset of the almost global section u. Denote the factor set M(Q, X )/∼ by C∞ (Q, X ). The set C∞ (Q, X ) can be naturally made a lattice-normed space. For instance, by the element u ~+~ v we mean the coset of the almost global section
q 7→ u(q) + v(q) q ∈ dom(u) ∩ dom(v) . If E is an order ideal in C∞ (Q) then we assign  E(X ) := u ∈ C∞ (Q, X ) : u ∈ E . In each coset u ~, there exists a unique section u ¯∈u ~ such that dom(v) ⊂ dom(¯ u) for all v ∈ u ~. The section u ¯ is called extended. The space C∞ (Q, X ) can thus be represented as the space of all extended almost global sections of the bundle X . The constant mapping X = Q × {X} that associates with each point of a topological space Q the same Banach space X is a simple example of a Banach bundle. If the totality C of constant functions c : Q → X is taken as a continuity structure C , then the continuous sections of the CBB (X , C ) are exactly continuous functions u : Q → X. Moreover, in this case we have C∞ (Q, X ) = C∞ (Q, X) and E(X ) = E(X). It will be shown in 2.4.8 that the space C(Q, X) is a Banach– Kantorovich space if and only if the trivial bundle Q × {X} is ample. 2.4.7. Theorem. Let X be a continuous Banach bundle over an extremal compact space Q and let E be an order-dense ideal in C∞ (Q). Then E(X ) is a Banach–Kantorovich space and C∞ (Q, X ) is its universal completion. C It is sufficient to establish that C∞ (Q, X ) is a universally complete Banach– Kantorovich space. Take a pairwise disjoint family (uξ )ξ∈„ in C∞ (Q, X ). Let Qξ be the least clopen set in Q on the complement of which uξ equals zero. Then Qξ ∩ Qη = ∅ for ξ 6= η. Without loss of generality we may assume that (Qξ )ξ∈„ is a partition of unity inSthe Boolean algebra of clopen subsets of Q. Assign Gξ := Qξ ∩ dom(uξ ) and G := ξ∈„ Gξ . Define a section u0 by the formulas dom(u) := G,

u0 |Gξ = uξ |Gξ (ξ ∈ „).

The set G is comeager due to 2.4.5, and the section u0 is continuous by definition. Consequently, u0 determines a unique element u ∈ C∞ (Q, X ) for which πξ uξ = πξ u

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(ξ ∈ „), where πξ is the projection induced by the operator of multiplication by P the characteristic function of the set Gξ . If θ is a finite set in „ and uθ = ξ∈θ uξ , then u − uθ ≤ πθ e, where πθ = sup{πξ : ξ ∈ θ}; therefore, u = bo-lim uθ . Now let a sequence (un )n∈N in C∞ (Q, X ) beTr-fundamental with regulator ∞ e ∈ C∞ (Q). On the comeager set G := dom(e) ∩ n=1 dom (un ), we may define a section u by the formula u(q) := lim un (q). Existence of the limits follows from the estimate kun (q) − um (q)k ≤ λk e(q) (n, m ≥ k), where λn → 0. After passing to the limit as m → ∞, it becomes clear from the last inequality that un converges to u locally uniformly. Consequently, the section u is continuous. It remains to observe that un − u ≤ λk e (n ≥ k), i.e., br-limn→∞ un = u. Thus, C∞ (Q, X ) is d-complete and r-complete and, therefore, the required assertion ensues from 2.2.3. B 2.4.8. A continuous Banach bundle X over an extremal compact space Q is called ample (or complete) if every bounded almost global continuous section of it can be extended to a global continuous section. Introduce the notation  C# (Q, X ) := u ∈ C∞ (Q, X ) : u ∈ C(Q) , i.e., C# (Q, X ) = E(X ) for E := C(Q). Let C(Q, X ) denote the set of all global continuous sections of X . The following assertions are equivalent: (1) the bundle X is ample; (2) C# (Q, X ) = C(Q, X ); (3) C(Q, X ) is a Banach–Kantorovich space. C The implications (1) ⇒ (2) ⇒ (3) are obvious. Suppose that (3) is satisfied and show that X is an ample bundle. Take a section u ∈ C# (Q, X ). There exist a partition of unity (Qξ )ξ∈„ in the Boolean algebra of clopen subsets of Q and a family of global continuous section (uξ )ξ∈„ such that ku(q) − uξ (q)kq ≤ 1/n for all q ∈ Qξ ∩ dom(u) and ξ ∈ „. This assertion can be easily deduced from 2.4.2 (5), since dom(u) is dense in Q. Let πξ be the projection in C∞ (Q, X ) that corresponds to the set Qξ . Due to bo-completeness of C(Q, X ), there exists a section vn ∈ C(Q, X ) with πξ vn = πξ uξ (ξ ∈ „). It follows from this that u − vn ≤ (1/n)1. Thus, the sequence (vn ) in C(Q, X ) converges relatively uniformly to u. Again we may apply bo-completeness of C(Q, X ) and conclude that u ∈ C(Q, X ), as required. B Let us show that assertions (1)–(3) are also equivalent to the following: (4) every bounded continuous section of the bundle X that is defined on a dense subset of Q can be extended to a global continuous section. It is not difficult to show that the trivial bundle Q × {X} is complete if and only if Q is a finite space or X is a finite-dimensional space.

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2.4.9. Let X and Y be continuous Banach bundles over the same compact space Q. A mapping h : q 7→ hq := h(q) ∈ L (Xq , Yq ) (q ∈ Q) is called a homomorphism from X into Y if hq is a bounded linear operator from Xq into Yq for all q ∈ Q, and, for
each continuous section u ∈ C(Q, X ), the mapping hu := h ⊗ u : q 7→ hq u(q) (q ∈ Q) is continuous. (1) If h is a homomorphism from X into Y then the function |||h||| : q 7→ kh(q)k is bounded. C For every q ∈ Q define a linear operator Tq : C(Q, X ) → Y (q) by setting Tq u = h(q)u(q). Endow the space C(Q, X ) with the uniform norm kuk∞ := sup{ku(q)kX (q) ; q ∈ Q} and observe that kTq k ≤ kh(q)k. Moreover, sup kTq uk = sup kh ⊗ uk < ∞ q∈Q

q∈Q

for every u ∈ C(Q, X ). We may apply the Uniform Boundedness Principle and derive supq∈Q kTq k < ∞. It remains to note that kh(q)k = sup{kh(q)xk : x ∈ X (q), kxk 6 1} = sup{kh(q)u(q)k : u ∈ C(Q, X ), kuk∞ 6 1} = kTq k. B We say that a homomorphism h is an isometry if hq is an isometry for each q ∈ Q. The set of all homomorphism from X into Y denote by HomQ (X , Y
). If h is an isometry of X onto Y then the mapping u 7→ hu u ∈ C(Q, X ) is a linear isometry of the lattice-normed spaces C(Q, X ) and C(Q, Y ). The converse assertion is also valid; however, a stronger assertion holds in the case of ample bundles. (2) Suppose that X and Y are ample Banach bundles over an extremal compact space Q. Then X and Y are isometric if and only if the LNSs C∞ (Q, X ) and C∞ (Q, Y ) are linearly isometric. C Let J be a linear isometry of C∞ (Q, X ) onto C∞ (Q, Y ). Then u ∈ C(Q, X ) if and only if Ju ∈ C(Q, Y ), since C = C# for
any ample bundle (see 2.4.8 (2)). Given u ∈ C(Q, X ) and q ∈ Q, assign hq u(q) := (Ju)(q). It is easy to see that
khq u(q) kq = k(Ju)(q)kq = Ju (q) = u (q) = ku(q)kq , i.e., hq is a linear isometry of Xq into Yq . Moreover, it is obvious that hu ∈ C(Q, Y ) whenever u ∈ C(Q, X ). For each y ∈ Yq , we may choose a global continuous section
−1 v so that v(q) = y. If u := J (v) then hq u(q) = y; hence, the operator hq is surjective. Consequently, h is an isometry of X onto Y . B

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2.4.10. Theorem. Every Banach–Kantorovich space X over an order-dense ideal E ⊂ C∞ (Q) is linearly isometric to E(X ) for some ample continuous Banach bundle X over Q. Moreover, such a bundle X is unique to within linear isometry. e be the universal completion of the space X (see 2.2.8). Assign C Let X  e : x ∈ C(Q) and kxkq := x (q) (q ∈ Q, x ∈ X0 ). It is clear X0 := x ∈
X that k · kq q∈Q is a multinorm in X0 . Let X be the continuous Banach bundle associated with the multinorm and let ı : X0 → C(Q, X ) be the canonical embedding (see 2.4.4). From the definition of the topology in X (see 2.4.3) it is clear that an arbitrary section u ∈ C(Q, X ) goes through the ε-tube defined by some section of the form ı(x), x ∈ X0 , on a suitable clopen neighborhood of each point of Q. Consequently, for every ε > 0, we may choose a partition of unity (Qξ )ξ∈„ in the Boolean algebra of clopen subsets of Q and a family (xξ )ξ∈„ of elements in X0 such that ku(q)−ı(xξ )(q)kq ≤ ε (q ∈ Qξ , ξ ∈ „); i.e., πξ u−πξ ı(xξ ) ≤ ε1 (ξ ∈ „), where πξ is the projection induced by the operator of multiplication by the characteristic P function of the set Qξ . In the space X0 there exists an element xε = πξ xξ , for which ı(xε ) − u ≤ ε1. Due to br-completeness of X0 , the limit x := br-lim xε also exists as ε → 0; moreover, ı(x) = u and x = u . Thus, ı(X0 ) = C(Q, X ), and, hence, C(Q, X ) is a Banach–Kantorovich space. According to 2.4.8, the bundle X is ample. The uniqueness assertion follows from 2.4.9. B 2.4.11. A continuous Banach bundle X is called a subbundle of a continuous Banach bundle X if X (q) is a Banach subspace of X (q) for every q ∈ Q and, moreover, C(Q, X ) = C(Q, X ) ∩ S(Q, X ). A subbundle X is called dense in X if every section u ∈ C(Q, X ) takes the values u(q) ∈ X (q) on a dense subset of Q. An ample hull or completion of a continuous Banach bundle X is any ample continuous Banach bundle X containing X as a dense subbundle. (1) Theorem. Every continuous Banach bundle over an extremal compact space has an ample hull unique to within isometry. C Consider a continuous Banach
bundle X over Q and endow the space C# (Q, X ) with the multinorm k · kq q∈Q , by writing kukq := u (q) for all u ∈ C# (Q, X ) and q ∈ Q. According to 2.4.4, the multinormed space thus obtained generates a continuous Banach bundle over Q that we denote by X . Let ı : C# (Q, X ) → C(Q, X ) be the corresponding canonical embedding. At each point q ∈ Q it is possible to define an isometric monomorphism h(q) : X (q) → X (q), satisfying the equality h(q)u(q) = ı(u)(q) for all u ∈ C(Q, X ). By Definition 2.4.9 it follows that h is a homomorphism from X into X and, thus, performs an isometry from X onto a Banach subbundle of X . According to above definitions, this subbundle is dense in X . Uniqueness for the completion of a continuous Banach bundle to within isometry follows from 2.4.9 (2). B

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(2) Theorem. Let X and Y be CBBs over Q, let X and Y be respective ample hulls, and let D be a comeager subset of Q. Then every homomorphism h ∈ HomD (X , Y ) has a unique extension h ∈ HomD (X , Y ). Moreover, the pointwise norms h and h coincide on a comeager subset of Q. ¯ ∈ X (q0 ) define C Take h ∈ HomD (X , Y ). For arbitrary q0 ∈ D and x h(q0 )¯ x ∈ Y (q0 ). To this end choose a section u ¯ ∈ C(Q, X ) such that u ¯(q0 ) = x ¯ and denote by u the restriction of u ¯ to comeager subset {q ∈ Q : u ¯(q) ∈ X (q)}. Then h ⊗ u is a continuous almost global section of Y . Let h ⊗ u ∈ C∞ (Q, Y ) be a maximal extension of h ⊗ u in Y . According to 2.4.9 (1) |||h||| is a locally bounded function. Therefore, |||H ⊗u||| is bounded on a neighborhood of q0 and, consequently, q0 ∈ dom(h ⊗ u). Now, we put h(q0 )x := h ⊗ u(q0 ). Clearly, h ⊗ u(q0 ) does not depend on the choice of the section u ¯ ∈ C(Q, X ) satisfying u ¯(q) = x ¯ and, hence, the desired homomorphism h is
correctly defined. The uniqueness of h follows from the equality H(q) extX (u)(q) = extY (H ⊗u)(q) for all u ∈ C# (Q, X ) and q ∈ Q. Since C∞ (Q, X ) is a universal completion of C(Q, X ), for every section u ∈ C∞ (Q, X ), u 6 1, one can find a net (uα )α∈A in C(Q, X ) bo-converging to u and satisfying the inequality uα 6 1. Then h ⊗ uα − h ⊗ u 6 |||h||| uα − u holds on D ∩dom(u), so that h⊗uα bo-converges to h⊗u . Thus, h⊗u 6 supα∈A h⊗uα and, for q ∈ D, we deduce |||h|||(q) = sup{ h ⊗ u (q) : u ∈ C(Q, X ), u 6 1}, ≥ sup{ h ⊗ u (q) : u ∈ C∞ (Q, X ), u 6 1} = |||h|||(q). The reverse inequality is obvious. B (3) Let X be a continuous Banach bundle, with X an ample hull of X . Then the spaces C∞ (Q, X ) and C∞ (Q, X ) coincide. C This follows from the definition of dense subbundle, since it is not difficult to check that a Banach bundle X is a dense subbundle of a continuous Banach bundle X if and only if every section u ∈ C∞ (Q, X ) takes the values u(q) ∈ X (q) on a comeager subset of Q. B 2.4.12. Now we introduce the continuous Banach bundle B(X , Y ) whose continuous sections are homomorphisms from a continuous Banach bundle X into a continuous Banach bundle Y . Consider a nonempty extremal compact space Q, and continuous Banach bundles X and Y over Q. (1) Let D
be a dense subset of Q, and let a mapping h : q ∈ D 7→ h(q) ∈ L X (q), Y (q) be such that hu ∈ C(D, Y ) for all u ∈ C(Q, X ). Suppose that the bundle X is ample. Then the pointwise norm |||h||| is continuous. In view of 2.4.2 (5), the equality |||h|||(q) = sup{hu(q) : u ∈ C(Q, X ), u 6 1}

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is valid for all q ∈ D which implies that the function |||h||| is lower semicontinuous. In order to prove upper semicontinuity for |||h||| at an arbitrary point q ∈ D we assume that |||h|||(q) < λ and establish the inequality |||h|||(p) < λ for all elements p in a neighborhood of q within D. Assume the contrary. Then, taking a number λ0 so that |||h|||(q) < λ0 < λ and defining V = {p ∈ D : |||h|||(p) > λ0 } we conclude that q ∈ cl V . For every element p ∈ V let a section up ∈ C(Q, X ) satisfy up 6 1 and hup (p) > λ0 . In view of continuity of the function hup at each point p ∈ V there is a clopen neighborhood Vp ⊂ Q such that S hup > λ0 on Vp ∩ D. It is easy to verify that the following equality holds: cl p∈V Vp = cl V . The Exhaustion Principle implies existence of a family (Up )p∈V of pairwiseSdisjoint clopen subsets of Q, satisfying the conditions Up ⊂ Vp (p ∈ V ) and cl p∈V Up = cl V . Since S X is ample, S the continuous bounded section p∈V up Up ∪ 0 Q\ cl V of X over the dense subset p∈V Up ∪ (Q\ cl V ) can be extended to a global section u ∈ C(Q, X ). Obviously, u 6 1. Since the sections u and up coincide on Up , it follows that hu > λ0 on cl V . The last assertion contradicts the inequality |||h|||(q) < λ0 . B (2) Every homomorphism h ∈ HomQ (X , Y ) has a continuous pointwise norm |||h||| provided that the bundle X is ample. The following assertion is a version of the Stone–Weierstrass Theorem for a CBB over an extremal compact space. The section coinciding with u on a clopen set V and vanishing on its complement is denoted by [V ]u. (3) Theorem. Let a vector subspace U ⊂ C(Q, X ) be stalkwise dense in X and contain [V ]u for all elements u ∈ U and clopen subsets K ⊂ Q. Then U is uniformly dense in C(Q, X ). C Suppose that a subspace U ⊂ C(Q, X ) meets the hypotheses of the theorem. We will fix a section v ∈ C(Q, X ) and, given an arbitrary ε > 0, construct an element u ∈ U satisfying the inequality ku − vk∞ ≤ ε. Given a point q ∈ Q choose a section uq ∈ U such that kuq (q) − v(q)k < ε. In addition, denote by Vq a clopen neighborhood of q on which uq − v < ε. Let us refine a finite subcover Vq1 , . . . , Vqn from the open cover (Vq )q∈Q of the compact Q. According to the Exhaustion Principle (see 1.1.6) there is a partition W1 , . . . , Wn of Q into clopen Pnsubsets such that Wk ⊂ Vqk for all k = 1, . . . , n. It is clear that the section u = k=1 [Wk ]uqk is the desired one. B 2.4.13. Theorem. Suppose that X is an ample bundle. Then there exists a (unique) continuous Banach bundle B(X , Y ) over Q such that the
stalk B(X , Y )(q) over
each point q ∈ Q is a Banach subspace of B X (q), Y (q) and C Q, B(X , Y ) = HomQ (X , Y ). C Consider the discrete Banach bundle over Q with the stalk over each point q ∈ Q equal to
the closure of the subspace {h(q) : h ∈ HomQ (X , Y )} in the bundle B X (q), Y (q) . In view of 2.4.12 (2), the space HomQ (X , Y ) is a continuity

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structure in this bundle and thus makes it into the desired continuous Banach bundle B(X , Y ). The equality
C Q, B(X , Y ) = HomQ (X , Y ) ensues from 2.4.12 (3). B 2.4.14. Theorem. Suppose that X is an ample bundle and D is a comeager subset of Q and U ⊂ C(D, X ) is
stalkwise dense in X on D. Let a mapping h : q ∈ D 7→ h(q) ∈ B X (q), Y (q)
be such that h ⊗ u ∈ C(D, Y ) for all elements u ∈ U . Then h ∈ C D, B(X , Y ) . C Let Y be an ample hull of Y . Using 2.4.11 (2), prove that there exists a section
h ∈ C∞ Q, B(X , Y ) such that the operator h(q) is an extension of h(q) for all q ∈ D ∩ dom(h). Moreover, h and h coincide on D ∩ dom(h). Assume that there exists an element ¯ For every q ∈ D := dom( h ) choose a section uq ∈ C(Q, X ) q0 ∈ D\ dom(h). and a clopen set Uq ⊂ Q such that q ∈ Uq and the inequalities uq 6 1 and h ⊗ uq > h
− ε hold on Uq . According to the Exhaustion Principle we may find a family Vq q∈D of pairwise disjoint clopen sets in Q such that Vq ⊂ Uq (q ∈ D) ¯ ∈ C(Q, X ) be a section coinciding with and the union of all Vq is dense in Q. Let u ¯ > h − 1. Since U is stalkwise dense in X on D there is an uq on Vq . Then h ⊗ u ¯(q0 )k < 1/2; thus, u − u ¯ 6 1/2 in a neighborhood element u ∈ U with ku(q0 ) − u of q and we deduce

>

kh(q0 )u(q0 )k = h ⊗ u (q0 ) = h ⊗ u (q0 )

1 ¯ (q0 ) > h⊗u − h · u−u h − 1 (q0 ) = ∞. 2

The contradiction shows that D ⊂ dom h. B 2.5. Measurable Banach Bundles In this section, we introduce the notion and establish some elementary properties of measurable sections obtained by means of a measurability structure and prove that every Banach–Kantorovich space over an ideal space is linearly isometric to the space of measurable sections of a measurable Banach bundle. 2.5.1. Throughout the section, (Š, †, µ) is a nonzero measure space. Let X be a Banach bundle over Š. Denote by S∼ (Š, X ) the set of all sections of X defined almost everywhere on Q. We call a set of sections C ⊂ S∼ (Š, X ) a measurability structure on X , if it satisfies the following three conditions:

80

Chapter 2 (a) λ1 c1 + λ2 c2 ∈ C for all λ1 , λ2 ∈ R and c1 , c2 ∈ C ; (b) the pointwise norm |||c||| : Š → R of every element c ∈ C is measur-

able; (c) the set C is stalkwise dense in X . If C is a measurability structure in X then we call the pair (X , C ) a measurable Banach bundle over Š. We shall usually write simply X instead of (X , C ) and denote the measurability structure C by CX . Let (X , C ) be a measurable Banach bundle over Š. We say that s ∈ S∼ (Š, X ) is a C -step-section (or simply Pna step-section, if it is clear which measurability structure is meant), if s = k=1 [Ak ]ck for some n ∈ N, A1 , . . . , An ∈ † and c1 , . . . , cn ∈ C . A section u ∈ S∼ (Š, X ) is called C -measurable (or simply measurable) if, for every K ∈ † with µ(K) < +∞, there is a sequence (sn )n∈N of C -step-sections such that sn (ω) → u(ω) for almost all ω ∈ K. The set of all C measurable sections of X is denoted by M (Š, †, µ, X |C ) or M (Š, X ) for brevity. 2.5.2. Suppose that X is a measurable Banach bundle over Š. We consider the equivalence relation ∼ in the set M (Š, X ) which is the coincidence almost everywhere: u ∼ v means that u(ω) = v(ω) for almost all ω ∈ Š. The coset containing an element u ∈ M (Š, X ) is denoted by u∼ . The factor set M (Š, X )/∼ is made into a vector space in the natural way: we write λu∼ + µv ∼ = (λu + µv)∼ for λ, µ ∈ R and u, v ∈ M (Š, X ). In addition, for every element u∼ ∈ M (Š, X )/∼ we may define its (vector)
∼ ∼ norm u := |||u||| ∈ M (Š). It is clear that the pair M (Š, X )/∼, · is an LNS over L0 (Š, †, µ); we denote it by L0 (Š, †, µ, X ), or shortly L0 (Š, X ). Note that the space L0 (Š, X ) can be endowed with the natural structure of a module over the ring L0 (Š, †, µ); as follows: e∼ u∼ := (eu)∼ for all e ∈ M (Š) and u ∈ M (Š, X ). 2.5.3. Theorem. If a measure space (Š, †, µ) possesses the direct sum property and if X is a measurable Banach bundle over Š, then L0 (Š, X ) is a Banach– Kantorovich space over L0 (Š, †, µ). C Decomposability of the LNS L0 (Š, X ) is obvious. In view of 2.2.3 for proving bo-completeness of L0 (Š, X ), it suffices to establish its d- and br-completeness. 0 Let (u∼ ξ )ξ∈„ be a family of elements of L (Š, X ), with pairwise disjoint supports Aξ ∈ B(Š) (see 1.4.11). Denote by „ the enrichment „ ∪ {∞} of the set „
∗ with a new element ∞ and define u∞ := 0 and A∞ := supξ∈„ Aξ . Fix a lifting ρ of L∞ (Š)S(see 1.4.9) and denote Aξ := ρ(Aξ ) for each ξ ∈ „. In view of 1.2.7 (2), the union ξ∈„ Aξ is measurable and differs from Š by a set of measure zero. It S is easy to verify that the section ξ∈„ uξ Aξ is measurable and the corresponding P 0 class is the desired sum bo- ξ∈„ u∼ ξ ∈ L (Š, X ).

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0 Now, suppose that a sequence (u∼ n )n∈N
in L (Š, X ) is br-fundamental. Then, for almost all ω ∈ Š the sequence un (ω) n∈N is fundamental. Due to completeness of the stalks of X , there exists a section u ∈ S∼ (Š, X ) to which the sequence (un )n∈N converges almost everywhere. It is clear that the section u is measurable and the corresponding class is the desired br-limit of the sequence (u∼ n )n∈N . B

2.5.4. Some important corollaries of the above theorem are now available. (1) Theorem. If (Š, †, µ) is a measure space possessing the direct sum property and E is an ideal of L0 (Š) then the set E(X ) := {u ∈ L0 (Š, X ) : u ∈ E} endowed with the operations induced from L0 (Š, X ) is a Banach–Kantorovich over E.  ∞ (2) The symbol L (Š, X ) stands for the set u ∈ M (Š, X ) : |||u||| ∈ ∞ L (Š) and its elements are called (essentially) bounded measurable sections of X . The totality of equivalence classes constituted by essentially bounded sections is denoted by L∞ (Š, X ). Obviously, the space L∞ (Š, X ) coincides with E(X ), where E = L∞ (Š, †, µ). In particular, L∞ (Š, X ) is a Banach–Kantorovich space over L∞ (Š, †, µ), (3) Let X be a Banach space. Consider the trivial Banach bundle X = Š × {X} and let the totality of constant functions c : Š → X be taken as the measurability structure of X . Then the set M (Š, X ) consists exactly of all Bochner measurable X-valued functions defined almost everywhere in Š: M (Š, X) = M (Š, X ). Consequently, L0 (Š, X) = L0 (Š, X ). In particular, L0 (Š, X), L∞ (Š, X), and E(X), where E is an order ideal in L0 (Š, †, µ), are Banach–Kantorovich spaces over L0 (Š, †, µ). 2.5.5. Let X be a measurable Banach bundle over Š. Consider a lifting ρ : L (Š) → L ∞ (Š) (see 1.4.8). We call a mapping ρX : L∞ (Š, X ) → L ∞ (Š, X ) a lifting of L∞ (Š, X ) associated with ρ if, for all u, v ∈ L∞ (Š, X ) and e ∈ L∞ (Š), the following relations hold: ∞

(1) ρX (u) ∈ u∼ and dom(ρX (u)) = Š; (2) |||ρX (u)||| = ρ( u ); (3) ρX (u + v) = ρX (u) + ρX (v); (4) ρX (eu) = ρ(e)ρX (u); (5) the set {ρX (u) : u ∈ L∞ (Š, X )} is stalkwise dense in X . In case there exist a lifting of L∞ (Š) and a lifting of associated with it, we say that X is a liftable measurable Banach bundle. 2.5.6. Let ρ be a lifting of L∞ (Š) and let X and Y be liftable measurable Banach bundles over Š. We call the bundles X and Y ρ-isometric, if their liftings

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ρX and ρY are associated with ρ and there exists an isometry h from X onto Y such that ρY (hu) = hρX (u) for all u ∈ L∞ (Š, X ). Two points ω1 , ω2 ∈ Š are said to be ρ-indistinguishable if ρ(f )(ω1 ) = ρ(f )(ω2 ) for every class f ∈ L∞ (Š). We say that a measurable Banach bundle X is invariant with respect to ρ, or ρ-invariant, if X (ω1 ) = X (ω2 ) and ρ(u)(ω1 ) = ρ(u)(ω2 ) for all u ∈ L∞ (Š, X ) for arbitrary ρ-indistinguishable points ω1 , ω2 ∈ Š. 2.5.7. The following assertion enables us to assume without loss of generality that every measurable Banach bundle under consideration is invariant with respect to the corresponding lifting. For every measurable Banach bundle over Š having a lifting associated with ρ, there is a ρ-stationary liftable measurable Banach bundle ρ-isometric to it. C Suppose that a measurable Banach bundle X meets the hypothesis of the assertion. Fix an arbitrary pair of Eρ-indistinguishable points ω1 , ω2 ∈ Š. It suffices to
construct an isometry ı from the stalk X (ω1 ) onto X (ω2 ) such that ı ρ(u)(ω1 ) = ρ(u)(ω2 ) for all u ∈ L∞ (Š, X ). Indeed, in this case, the stalks over indistinguishable points can be identified by means of such isometries. For each ω ∈ Š, denote by X0 (ω) the subspace {ρ(u)(ω) : u ∈ L∞ (Š, X )} of the stalk X (ω). Let classes u, v ∈ L∞ (Š, X ) be such that ρ(u)(ω1 ) = ρ(v)(ω1 ). Then ρ(u)(ω2 ) = ρ(v)(ω2 ), because from the properties of lifting it follows that kρ(u)(ω2 )−ρ(v)(ω2 )k = ρ( u−v )(ω2 ) = ρ( u−v )(ω1 ) = kρ(u)(ω1 )−ρ(v)(ω1 )k = 0. This enables
us to consider a bijection ı0 : X0 (ω1 ) → X0 (ω2 ), defined by the rule ı0 ρ(u)(ω1 ) = ρ(u)(ω2 ) for every class u ∈ L∞ (Š, X ). Due to the fact that the subspaces X0 (ω) are dense in the corresponding stalks X (ω), the isometry ı0 can be extended to the desired isometry ı : X (ω1 ) → X (ω2 ). B 2.5.8. Suppose that Q is the Stone space of the Boolean algebra B(Š) and τ : Š → Q is the canonical immersion of Š into Q corresponding to the lifting ρ of L∞ (Š). Let Y be an ample continuous Banach bundle over Q and X = Y ◦ τ . If C is a continuous structure in Y , then the set C ◦ τ is a measurability structure in X , since |||c ◦ τ ||| = |||c||| ◦ τ and |||c||| ◦ τ is a measurable function. The bundle Y ◦ τ is always regarded as a measurable Banach bundle with respect to the measurability structure C ◦ τ . For every v ∈ C∞ (Q, Y ) the composite v ◦ τ is a measurable section of the bundle X . C This is evident. B For v ∈ C∞ (Q, Y ) we denote by (v ◦ τ )∼ the coset containing an element v ◦ τ .

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2.5.9. Theorem. The mapping v 7→ (v ◦ τ )∼ is an isometry from the BKS,
C∞ (Q, Y ) onto L0 (Š, X ) and is associated with the isomorphism e 7→ (e ◦ τ )∼ : C∞ (Q) → L0 (Š). The image of C(Q, Y ) under this isometry is L∞ (Š, X ). C We need only to prove that an almost everywhere defined section u of the bundle X is measurable if and only if u ∼ v ◦ τ for some element v ∈ C∞ (Q, Y ). In other words, it should be verified that U = L 0 (Š, X ), where
U := {u ∈ S∼ (Š, X ) : ∃v ∈ C∞ (Q, Y ) u ∼ v ◦ τ }. We confine ourself to the case of a σ-finite measure space. In view of 2.5.8 U ⊂ L 0 (Š, X ). Let us prove the reverse inclusion. ^ := A ^∼ be the clopen set in Q corresponding Let v ∈ C∞ (Q, Y ), A ∈ † and A to A under the Stone transform. Then 

^ (v ◦ τ ) = [A]v ^ ◦ τ, [A](v ◦ τ ) ∼ [ρ(A∼ )](v ◦ τ ) = τ −1 A which implies that the set U contains the fragments of all its elements. Thus, U contains also the set of all C -step-sections. Suppose that u ∈ L 0 (Š, X ). By the definition of measurability in 2.5.1, there exists a sequence of step-sections un ∼ vn ◦ τ that converges
to a section u ∈ S∼ (Š, X ) almost everywhere. Then we ∼ ∼ 0 have o-limn→∞ |||un − u||| = o-limu∼ n − u = 0 in the K-space L (Š, †, µ); thus, ∼ ∼ u∼ is an order isomorphism n − um → 0. Since the correspondence f 7→ (f ◦ τ ) 0 of vector lattices L (Š, †, µ) and C∞ (Q), it follows that o-limn→∞ vn − vm = 0 in C∞ (Q). Due to o-completeness of the LNS C∞ (Q, Y ), there exists the o-limit v ∈ C∞ (Q, Y ) of the sequence (vn ). Obviously, u ∼ v ◦ τ . B The inverse isometry from L0 (Š, X ) onto C∞ (Q, Y ) is defined by the rule ^, where u ^ is a unique section in C∞ (Q, Y ), representing u as (^ u ◦ τ )∼ . We u 7→ u ^ as the Stone transform of u. refer to the section u ^ ◦ τ is a lifting of L∞ (Š, X ) associated with ρ. Endowed The mapping u 7→ u with this lifting, the measurable Banach bundle X is ρ-stable. Theorem 2.5.9 describes a method of constructing a liftable measurable Banach bundle given a complete continuous Banach bundle over the corresponding Stone space. The following result shows that every liftable measurable Banach bundle can be obtained exactly in such a way. 2.5.10. Theorem. Let X be a ρ-invariant measurable Banach bundle over Š that has a lifting associated with ρ. Then there exists an ample continuous Banach bundle X^ over Q unique to within an isometry and such that X = X^ ◦ τ and ρ(u) = u ^ ◦ τ for all u ∈ L∞ (Š, X ). C Since X
is ρ-stationary, we may define a Banach bundle Y overτ (Š) by the formula Y τ (ω) := X (ω) and endow it with the continuity structure ρ(u) ◦ τ −1 : u ∈ L∞ (Š, X ) .

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Denote by U the vector space C b (D, Y ) with the multinorm k·kq q∈Q , kukq = u (q), where u : Q → R is a unique continuous extension of the bounded function q 7→ ku(q)k (q ∈ D). Let X^ be the CBB over Q generated by the multinormed space U and let ı : U → C(Q, X^) be corresponding canonical embedding (see 2.4.4). Since U is norm complete, stalkwise dense in X^ and invariant with respect to band projections, we may prove that ı(U ) = C(Q, X^). For each point q ∈ D we may define a linear isometry h(q) between the stalks X (q) and X^(q) satisfying the equality h(q)u(q) = ı(u)(q) for all u ∈ U . It is easy to see that h is an isometry ^ from X onto X D . Consequently, X^ is the desired continuous Banach bundle. Most of the necessary properties of the bundle X^ are easily verified. Prove that the bundle is ample. It follows from the construction of the bundle X^
that for each class u ∈ ∞ L (Š, X ), the bounded section ρ(u) ◦ τ ∈ C τ (Š), Y can be extended to an element of C(Q, X^) that will be denoted by u ^. It can be easily verified that the ^ is an isometry from the LNS L∞ (Š, X ) onto C(Q, X^) associated mapping u 7→ u e) from L∞ (Š) onto C(Q), see 1.4.9. It follows that with the isomorphism (e 7→ ^ the image of L∞ (Š, X ) under this isometry is o-dense in C(Q, X^). In view of the o-completeness of L∞ (Š, X ), this image coincides with C(Q, X^) and, hence, the LNS C(Q, X^) is o-complete as well. Uniqueness of the bundle X^ follows from 2.4.10. B We call the ample continuous Banach bundle X^ in the statement of the last theorem the Stone transform of the measurable Banach bundle X . Three facts presented below are immediate consequences of applying Theorem 2.5.9 and the representation result 2.4.9. 2.5.11. Theorem. Every Banach–Kantorovich space X over an order-dense ideal E ⊂ L0 (Š) is linearly isometric to E(X ) for some liftable measurable Banach bundle X over Š. Moreover, such a bundle X is unique to within a linear isometry. 2.5.12. Theorem. For every Banach–Kantorovich X over E and every isomorphism ı from the K-space E onto an order-dense ideal F ⊂ L0 (Š) there exist a liftable measurable Banach bundle X over Š (unique to within isometry) and some isometry from X onto F (X ) associated with ı. 2.5.13. Theorem. For every Banach–Kantorovich space X over a Kantorovich–Pinsker space, there exist a measurability structure Š, possessing the direct sum property, an order-dense ideal F ⊂ L0 (Š), and a liftable measurable Banach bundle X over Š such that the LNSs X and F (X ) are isometric. 2.6. Comments 2.6.1. (1) The concept of lattice-normed space was introduced for the first

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time by L. V. Kantorovich in 1936 [154]. These are vector spaces normed by elements of a vector lattice. Somewhat earlier, G. Kurepa [192] considered “espaces pseudodistanci´es,” i.e. a space with a metric that takes values in an ordered vector space. First applications of vector norms and metrics were related to the method of successive approximations in numerical analysis, see [154, 161, 163, 182, 340]. (2) It is worth stressing that [157] is the very paper in which the unusual decomposability axiom (see 2.1.1 (4)) for an abstract norm appeared for the first. Paradoxically, this axiom (4) was often omitted as inessential in the further research by other authors. The profound importance of (4) was rediscovered in connection with Boolean-valued analysis (see [196, 197]). (3) The connection between decomposability and existence of a Boolean algebra of projections in an LNS was discovered by A. G. Kusraev [196, 197]. Spaces with a fixed Boolean algebra of linear projections and a coordinated order (the so-called coordinated spaces) were studied by J. L. B. Cooper [69, 70]. Assertions 2.1.7 (1, 3) were obtained in [181]. (4) The notion of discrete element is important in the structure theory of vector lattices, see [163, 263, 388]. Discrete functionals and discrete (real-valued) measures are well studied and have a simple structure [139, 326]. As was shown by J. A. Crenshaw [71], under certain mild conditions a discrete element in the lattice of order-bounded operators is completely defined by a discrete functional on the domain vector lattice and a discrete element of the target vector lattice. Thus discrete operators comprise a poor class. At the same time there is a series of interesting results in which the concept of module discreteness plays the central role (see for instance [42, 197, 215, 220, 344, 393]). This motivates the study of concepts analogous to module discreteness, module atomicity, and module indecomposability in lattice-normed spaces. The notion of norm-n-indecomposable element from 2.1.9 and Theorem 2.1.10 are due to V. A. Radnaev [322, 323]. 2.6.2. (1) The completeness criterion 2.2.3 was stated by A. G. Kusraev in [196] under the condition that the norm lattice E is order complete. In [197], this was proven in a more general situation of spaces with decomposable vector multinorms. The assumption of order completeness for E was removed in [181]. For an Archimedean vector lattice (the case of X = E) the indicated fact was established by A. I. Veksler and V. A. Ge˘ıler [381]. (2) The concept of universal completion (maximal extension) for an arbitrary K-space was introduced and studied by A. G. Pinsker; see [163]. He established, in particular, that any K-space has a unique, to within an isomorphism, maximal extension. Assertion 2.2.8, which is a generalization of the Pinsker Theorem for LNSs, was essentially obtained in [197]. As regards Theorem 2.2.11 (1) on

86

Chapter 2

^ = oX in 2.2.11 (2) is order completion of an LNS, see [197, 218]. The assertion X due to A. E. Gutman. The properties of approximating sets and their applications to order-bounded operators were studied in [120]. (3) The operator-dual space X ∗ in 2.2.4, introduced here for constructing the maximal extension mX, is of independent interest; it was studied in [197]. In particular, an LNS X is operator-reflexive in the sense that X ∗∗ = κ(X) if and only if the set {x ∈ X : x ≤ e} is weakly cyclically compact for each element e ∈ E+ (see [197]). More about the conception of cyclical compactness will be said in Section 8.5. 2.6.3. (1) Information about measurable functions with values in a Banach space and, particularly, in the space of bounded linear operators, is presented in [74, 78, 87]. A general idea of constructing spaces in Section 1.2 can be expressed as follows: If X is a Banach (or locally convex) space and E is a function space then we may associate with them a class Z of vector-functions (measurable or continuous) by requiring that f ∈ Z if and only if l ◦f ∈ E for each l ∈ X 0 , etc. (see [54, 92, 238, 319]). This idea was elaborated while developing the theory of vector integration [51, 74, 78, 87, 91, 375]. (2) From 2.3.4 it is clear that the algebraic tensor product E ⊗ X is bo-dense in the space E(X). Denseness of E ⊗ X in E(X) with respect to the scalar norm is connected with order continuity of a Banach lattice E: see Phuong-Cac [319], A. V. Bukhvalov [55], and V. L. Levin [237, 238]. The vector norm in E ⊗ X, that is introduced in 1.2.10, is analogous to Levin’s cross-norm, see [237, 238]. Assertion 2.3.4 (1) is proven as in [238, Theorem 4.2]. (3) The tensor product of Banach lattice E and an arbitrary Banach space with cross-norm induced by the space of regular operators was introduced and learned by V. L. Levin [238]. V. T. Khudalov [169, 170] studied a similar crossnorm for a Banach space with regular cone. An important method for constructing tensor products of vector and Banach lattices was proposed by D. Fremlin [98, 99]. G. Shotaev [350, 351] constructed tensor products of lattice-normed spaces and of Banach spaces with mixed norm. (4) In the study of measurable vector-functions the following question is of interest: If X = F is a Banach ideal space of functions of a variable s then every vector-function ~f : T → X = F generates a function of two variables by the formula φ(s, t) = (~f (t))(s). However, φ(·, ·) may fail to be measurable as a function of two variables even in the simplest cases (for instance, for the measure space [0, 1] with the Lebesgue measure and X = L2 (0, 1)). Indeed, W. Sierpinski constructed an example of a subset of a square which is Lebesgue nonmeasurable and has at most two common points with every straight line. The characteristic function φ0 of this set is nonmeasurable as a function of two variables whereas the corresponding

Lattice-Normed Spaces

87

vector-function f is the zero function since φ0 (s, t) = 0 almost everywhere for all t. By adding φ0 to nonzero measurable vector-functions, we obtain similar examples with nonzero vector-functions. (5) The fact that Property (C) in F implies measurability for the function K (t) := kK(·, t)kF (see 2.3.9 (1)) was established by W. A. J. Luxemburg. It can be also deduced from one result of Yu. I. Gribanov [115]; see [162, XI.1.3, XI.1.4]. Proposition 2.3.9 (3) is essentially due to H. W. Ellis [90]. The case of an atomic measure is of no avail since then all functions are measurable. (6) A much more difficult question is the representation f (t)(·) = K(·, t) for weakly measurable functions. One of the possible solutions to this problem rests on the Bukhvalov integrality criterion 6.5.4 (see [59, 228]). If F is a Banach ideal space with order continuous norm then E(F 0 ) = Es (F 0 ). 2.6.4. (1) The invention of Banach bundles is customarily connected with the name of J. von Neumann who proposed in 1937 some ideas about varying Banach spaces. The corresponding formal descriptions appeared about 1950 in the papers of R. Godement, I. Kaplansky, and I. M. Gelfand and M. A. Na˘ımark. Presently, the theory of continuous Banach bundles is a rather wide area of research. An adequate description for the current state of this theory can be found in the surveys and references of [95]. The monograph [104] contains a detailed discussion of most of the necessary notions related to CBBs. Continuous Banach bundles are often used for representing various functional-analytical objects, see [95, 104, 131, 339]. (2) In the article [218] by A. G. Kusraev and V. Z. Strizhevski˘ı, the space E(X ) was introduced of almost global sections of a continuous Banach bundle X , and Theorem 2.4.7 was established; it was also shown that each BKS is linearly isometric to E(X ) for a suitable X; however, uniqueness of the bundle X was not established (cf. the existence result in Theorem 2.4.10). A. E. Gutman found a class of uniqueness for Theorem 1.4.7, the class of ample continuous Banach bundles, and proved Theorem 1.4.10 [123]. For a detailed presentation of these and other interesting results see [118–124]. The material of Section 2.4 shows that ample CBBs have other advantages. For instance, only after assuming the completeness of a CBB X it becomes possible to introduce the CBB L (X , Y ) and the dual CBB X 0 (2.4.13); this result is also due to A. E. Gutman [118]. 2.6.5. (1) The idea of a measurability structure has been proposed by N. Dinculeanu as early as in 1966, but has not been much studied since then. A different approach to defining measurability of sections has been prevalent so far in the papers on Banach bundles. Namely, let X be a continuous Banach bundle over a locally compact space Š with a fixed Radon measure. A section u ∈ S(Š, X ) is called measurable if, for every compact K ⊂ Š, there exists a sequence of continuous sections un ∈ S(Š, X ) converging to u almost everywhere on K.

88

Chapter 2

(2) In Section 2.5 we follow A. E. Gutman [119]. The way of introducing measurable sections in Section 2.5 is similar to Daniel’s construction and is formally more general than the traditional topological approach. As is seen from 1.5.10, liftable measurable Banach bundles in the class of all MBBs occupy in a sense the same place as complete continuous Banach bundles in the class of all CBBs. The connection between liftable measurable Banach bundles and complete CBBs, described in 2.5.8–2.5.10, enables us to transfer the facts of the theory of complete CBBs to the case of a measurable Banach bundle; see [119].

Chapter 3 Positive Operators

In this chapter we briefly present some basic results on positive operators in vector lattices. Two natural classes of operators are closely related to positive operators: the set of differences of positive operators called the space of regular operators, and the space of order-bounded operators (3.1.1). The fundamental Riesz–Kantorovich Theorem (3.1.2) claims that, under natural conditions, the space of regular operators is an order complete vector lattice and that a regular operator can be described as an order-bounded operator, i.e., an operator carrying orderbounded sets into order-bounded sets. Explicit expressions for finite and infinite lattice operations as well as for the moduli, positive and negative parts of orderbounded operators comprise the so-called order calculus. One of the approaches leans upon the concept of a generating set of projections (3.1.5). All these facts require order completeness of the target vector lattice. Order complete vector lattices appear by necessity also in the problem of dominated extension of linear operators: such an extension is possible in a mass setting if and only if the target vector lattice is order complete (the Hahn–Banach–Kantorovich Theorem (3.1.7) and the Bonnice–Silvermann–To Theorem (3.1.8)). Simplest representatives of the class of order-bounded operators are orthomorphisms (3.3.2) and lattice homomorphisms (3.3.1). Hahn–Banach–Kantorovichtype theorems are also true for lattice homomorphisms (3.3.10, 3.3.11). This fact is closely connected with the extreme structure of convex sets of operators (3.3.7– 3.3.9). Among order bounded operators are important classes of order continuous and order σ-continuous operators (3.2.1). Their antipodes, called singular and σsingular operators, are distinguish by the property of vanishing on an order-dense or σ-order-dense ideal, respectively (3.2.1). Order continuous and order σ-continuous operators comprise bands (3.2.3 (2)). Moreover, these bands are the disjoint complements of the sets of singular and σ-singular bands under the appropriate conditions (3.2.3 (1)). It is a long tradition stemming from measure theory to look for some

90

Chapter 3

explicit description of different fragments of an operator or measure. Several results in this direction are presented in 3.2.5–3.2.9 and 3.3.13. Next we consider the class of the so-called Maharam operators. A Maharam operator is a “full-valued” or order interval preserving order continuous operator. One of the main peculiarities of a Maharam operator is that the Boolean algebra of its fragments is isomorphic to the base of the domain vector lattice (3.4.5). Variants of the Hahn Decomposition Property and the Radon–Nikod´ ym Theorem are valid for Maharam operators (3.4.6 (1), 3.4.9). An important example of a Maharam operator is the transformation assigning to each order-bounded operator its restriction to a fixed massive sublattice of the domain vector lattice (3.5.2 (6)). This fact enables us to prove the existence of simultaneous extension from an arbitrary massive sublattice (3.4.11). Maharam operators may be useful in studying different classes of operators. Given a positive operator, we may always extend it to a Maharam operator by a construction very much resembling the Daniell construction of the Lebesgue integral (3.5.2, 5.5.3). The domain of the so-extended operator is then approximated in a way by the domain vector lattice of the initial operator (3.5.6–3.5.8). The band generated by the initial operator is linear and order isomorphic to that of the resulting Maharam operators (3.5.5). These results are used to obtain some description for the Boolean algebra of fragments of a positive operator by up-and-down procedures (3.5.9, 3.5.10). 3.1. Operators in Vector Lattices In this section we introduce spaces of order-bounded operators and present several fundamental results which are systematically used in the sequel. 3.1.1. Let E and F be vector lattices. A linear operator T : E → F is said to be positive provided that T (E+ ) ⊂ F+ ; regular provided that it can be represented as a difference of two positive operators; last, order-bounded or o-bounded provided that T carries every order-bounded subset in E onto an order-bounded subset in F . The set of all regular, order-bounded, and positive operators from E into F is denoted by Lr (E, F ), L∼ (E, F ), and L+ (E, F ) := L∼ (E, F )+ , respectively. The classes Lr (E, F ) and L∼ (E, F ) are vector subspaces in the vector space L(E, F ) of all linear operators from E to F , provided with the order relation: S ≥ T ⇔ S − T ≥ 0. Every positive operator is obviously order-bounded and the difference of orderbounded operators is order-bounded, too. Thus, every regular operator is orderbounded. The converse may fail but is also true in the case of order complete F . The latter follows immediately from the famous Riesz–Kantorovich Theorem to be proved in the next subsection. First we give a simple auxiliary fact.

Positive Operators

91

Let E be a vector lattice, let X be an arbitrary real vector space, and let U be an additive and positively homogeneous mapping from E+ to X: U (x + y) = U x + U y,

U (λx) = λU x (0 ≤ λ ∈ R; x, y ∈ E+ ).

Then U admits a unique linear extension T to the whole vector lattice E. If in addition F is a vector lattice and U (E+ ) ⊂ F+ then T is positive. C The uniqueness of extension is obvious from the formula x = x+ − x− ; therefore, it remains to prove existence. For every x ∈ E, we put T x = U x+ − U x− and observe that T is the sought extension. Indeed, for z = x − y, x, y ∈ E+ we have z + − z − = x − y or z + + y = x + z − . Therefore, U z + + U y = U x + U z − by hypotheses, whence T z = U z + − U z − = U x − U y = T x − T y. Now, we may conclude that T is additive on E. Additivity implies that T (−x) = −T x for all x ∈ E, which involves homogeneity of T by virtue of its positive homogeneity. B 3.1.2. Riesz–Kantorovich Theorem. If E is a vector lattice and F is some K-space then the set of order-bounded operators L∼ (E, F ) ordered by the cone of positive operators L∼ (E, F )+ is a K-space. Moreover, for every S, T ∈ L∼ (E, F ) and x ∈ E+ , the following formulas hold: (1) (S ∨ T )x = sup{Sx1 + T x2 : x1 , x2 > 0, x = x1 + x2 }; (2) (S ∧ T )x = inf{Sx1 + T x2 : x1 , x2 > 0, x = x1 + x2 }; (3) S + x = sup{Sy : 0 ≤ y ≤ x}; (4) S − x = − inf{Sy : 0 ≤ y ≤ x}; (5) |S|x = sup{|Sy| : |y| ≤ x}; Pn Pn (6) |S|x = sup { i=1 |Sxi | : x1 , . . . , xn > 0, x = i=1 xi , n ∈ N} ; (7) |Sx| ≤ |S|(|x|) (x ∈ E). C To establish that some order vector space L is a vector lattice, it suffices to check that, for every x ∈ L, there exists |x| = x ∨ (−x). Then it remains to use the formulas 1.3.2 (3). Denote by U x the right-hand side of the formula (5) and prove that the operator U : E+ → F is additive and positively homogeneous as in 2.1.1. Observe that for a fixed x ∈ E+ the set {y ∈ E : |y| ≤ x} is bounded and, since S ∈ L∼ (E, F ), its image {|Sy| : |y| ≤ x} is also bounded in F . Thus, the supremum in (5) exists and the operator U is correctly defined. Positive homogeneity of U is obvious and it remains to prove its additivity. Formula (5) can be slightly simplified. By virtue of associativity of bounds, we have U x = sup{|Sy| : |y| ≤ x} = sup{(Sy) ∨ (S(−y)) : |y| ≤ x} = sup{Sy : |y| ≤ x}.

92

Chapter 3

Take now x1 , x2 ∈ E+ and prove that U (x1 + x2 ) = U x1 + U x2 . If y1 and y2 are such that |y1 | ≤ x1 and |y2 | ≤ x2 then |y1 + y2 | ≤ x1 + x2 and Sy1 + Sy2 = S(y1 + y2 ) ≤ U (x1 + x2 ), whence U x1 + U x2 ≤ U (x1 + x2 ). Prove the reverse inequality. Let |y| ≤ x1 + x2 . Put y1 := x1 ∧ y + − x1 ∧ y − , y2 := y − y1 . It is easy to check that |y1 | ≤ x1 , |y2 | ≤ x2 , whence |Sy| = |S(y1 + y2 )| ≤ |Sy1 | + |Sy2 | ≤ U x1 + U x2 which proves additivity. In view of 3.1.1, the operator U admits a unique linear extension to E which we denote by |S|. Check that |S| = S∨(−S). From a simplified version of the formula for U it is obvious that [S] ≥ ±S and that V ≥ ±S yields V ≥ [S] by which the claim is verified. Thus, we established that L∼ (E, F ) is a vector lattice and the modulus of an operator in the vector lattice can be calculated by formula (5). Formulas (1)– (4) and (7) easily follow from (5) as was mentioned above. Prove that the vector lattice L∼ (E, F ) is order complete. To this end it suffices to prove existence of a least upper bound of an arbitrary bounded above set. Without loss of generality, we may assume that the set under study is directed upward. Thus, let M be an upward-directed set of operators and let V ∈ L∼ (E, F ) be an upper bound of it. Given x ∈ E+ , we put T x := sup{Sx : S ∈ M }. The supremum exists since Sx ≤ V x for all S ∈ M and all x ∈ E+ . Again we may apply the proposition from 3.1.1 to the operator U . Let T be an extension of U to E. Then T belongs to L∼ (E, F ), since S ≤ T ≤ V for every S ∈ M . It is now seen that T = sup M . Thus, we verified that L∼ (E, F ) is order complete. It remains to prove the formula (6). According to (7), the right-hand side in (6), say f , is less or equal to the left-hand one. To prove the reverse inequality fix x ∈ E+ and take y, |y| ≤ x. Then we have Sy = Sy + − Sy − ≤ |Sy + | + |Sy − | + |S(x − |y|)| ≤ f and the claim easily follows. B 3.1.3. Let E and F be vector lattices with F order complete. Assume that given a positive operator S : G → F on an order ideal G ⊂ E. If the set S([0, e]∩G) is order-bounded in F for every e ∈ E+ then we may define EG (S)e := sup{Sg : g ∈ G, 0 ≤ g ≤ e} := sup{S(g ∧ e) : g ∈ G} (e ∈ E+ ).

Positive Operators

93

An operator EG (S) : E+ → F is additive and positively homogeneous, so it can be extended to E by differences, see 3.1.1. The resulting operator, called the minimal extension of S, is denoted by the same symbol E (S) := EG (S). Specify some properties of minimal extensions. (1) The minimal extension E (S) agrees with S on G and vanishes on G⊥ . C It follows directly from the definition of minimal extension. B Let Lext (G, F ) denote the set of regular operator S ∈ L∼ (G, F ) such that S + and S − admit minimal extension. It is evident that S ∈ Lext (G, F ) if and only if there is a positive operator Se ∈ L∼ (E, F ), a dominant for S, such that e |S|(g) ≤ S(g) (g ∈ G+ ). Since the operator EG defined on Lext (G, F )+ is additive and positively homogeneous, it can be extended by differences as in 3.1.1 and we obtain a positive operator EG : Lext (G, F ) → L∼ (E, F ) which is called the minimal extension operator. Let RG : L∼ (E, F ) → L∼ (G, F ) be the restriction operator T 7→ T |G . The restriction operator is also positive. (2) The minimal extension operator EG preserves order continuity and sequential order continuity. C Ensure that EG preserves sequential order continuity. Take 0 < S ∈ Lext nσ (G, F ) and denote Se := EG S. For any increasing sequence (en )n∈N in E+ with supn en = e we have e = sup{S(g ∧ e) : g ∈ G} = sup{sup S(g ∧ en ) : g ∈ G} S(e) n

e n ). = sup sup{S(g ∧ en : g ∈ G} = sup S(e n

n

e n ) % S(e). e e 1 − en ) % S(e e 1 ), Thus, S(e Now, the relation en & 0 implies that S(e e n ) & 0. B whence S(e The null ideal of an operator T ∈ L∼ (E, F ) is the set N (U ) = { x ∈ E : U (|x|) = 0}. The disjoint complement N (U )⊥ is called the carrier or the band of essential positivity of the operator T and is denoted by CT . In case when N (U )⊥ = E we say that U is essentially positive. (3) The operator πG := EG ◦ RG is a band projection in L∼ (E, F ) and ⊥ ∼ πG L (E, F ) = {T ∈ L∼ (E, F ) : G ⊂ N (T )}.

C It is sufficient to prove that S := πG T is a fragment of T for every positive T ∈ L∼ (E, F ). Let V := S ∧ (T − S). If x ∈ E+ then Sxα % Sx for some increasing net (xα ) ⊂ G according to the definition of πG . Since V ≤ S, we have V (x − xα ) ≤ S(x − xα ) & 0. In virtue of (1) V vanishes on G; therefore, V x = o-lim V xα = 0. B

94

Chapter 3

(4) The subspace Lext (G, F ) is a band in L∼ (G, F ) linearly and latticially isomorphic to πG L∼ (E, F ) under the minimal extension operator. 3.1.4. Thus, we have constructed an isotonic mapping G 7→ πG from the lattice I (E) to the Boolean algebra P(L∼ (E, F )). Now, consider some properties of this mapping. In case G = e⊥⊥ we put π[e] := πG . (1) Let π be a band projection in E and G = π(E). Then πG T = T ◦ π for every T ∈ L∼ (E, F ). In particular, π[e] T = T ◦ [e], where [e] stands for the band projection onto {e}⊥⊥ . C According to the definition of minimal extension πG T x = sup{T (π(g ∧ x)) : g ∈ G} = sup{T (g ∧ π(x)) : g ∈ G} = T (πx). B (2) Suppose that C and D are order ideals in E and G = C ∩ D. Then πG = πC ∧ πD . In particular, πG and πD are disjoint if and only if C ∩ D = {0}. C Since C ∩ D = {c ∧ d : c ∈ C, d ∈ D}, for 0 ≤ x ∈ E and 0 ≤ T ∈ L∼ (E, F ) we deduce πG T (x) = sup{T (x ∧ c ∧ d) : c ∈ C, d ∈ D} = sup sup{T (x ∧ c ∧ d) : d ∈ D} = πC (πD T )(x). c∈C

Thus, πG = πC ◦ πD = πC ∧ πD . B (3) Suppose that C and D are order ideals in E and G = C + D. Then πG = πC ∨ πD . C Monotonicity of G 7→ πG implies πG ≥ πC ∨ πD . If u ∈ [0, x] ∩ (C + D) then u = c + d for some c ∈ C+ and d ∈ D+ and for x1 := c, x2 := x − c we have x = x1 + x2 , c ∈ [0, x1 ] ∩ C, and d ∈ [0, x2 ] ∩ D. Therefore, T u = T c + T d ≤ πC T (x1 ) + πD T (x2 ) ≤ (πD T ∨ πD T )x ≤ (πD ∨ πD )T x and πG T (x) = sup T u ≤ (πD ∨ πD )T x. B S (4) Let (Gα ) be an increasing family of order ideals in E and G := Gα . Then G is also an order ideal and πG = supα πGα . Moreover, the representation holds: ⊥ ∼ πG L (E, F ) = {T ∈ L∼ (E, F ) : (∃α) Gα ⊂ N (T )}. C This follows immediately from the definitions on using associativity of least upper bounds: (πG T )x = sup T (g ∧ x) = sup sup T (g ∧ x) = (sup πGα T )x. B g∈G

α g∈Gα

α

If G := E(e) is the ideal generated by e ∈ E+ then we write πe instead of πG .

Positive Operators

95

(5) The following representation for πe is valid: πe T x = sup T (ne ∧ x)

(x ∈ E + , T ∈ L+ (E, F )),

n

πe T x = πe T x+ − πe T x− πe T = πe T + − πe T −

(x ∈ E, T ∈ L+ (E, F )), (T ∈ L∼ (E, F )).

3.1.5. Very often it is useful to have formulas for the calculation of (T ∨S)x and (T ∧ S)x using the fragments of T and S. Different approaches to this problem can be unified by means of the following notion. A set of projections P ⊂ P(L∼ (E, F )) is said to be generating if for all T ∈ L+ (E, F ) and x ∈ E we have T x+ = sup{pT x : p ∈ P}. As an easy example we cite the following. To each band projection π ∈ P(E) assign π T 7→ T ◦ π acting in L∼ (E, F ) and denoted by P ◦ the set of the band projection ^ all such band projections. (1) Put P π := {πe : e ∈ E+ }. Then P π is a generating set of projec∼ tions in L (E, F ). C Indeed, if e := x+ then πe T x+ = T x+ and πe T x− = 0 by 3.1.3 (1); therefore πe T x = πe T x+ − πe T x− = T x+ . B (2) If a vector lattice E has the strong Freudenthal property then P ◦ is a generating set of projections in L∼ (E, F ). C According to 1.3.9 (2) the strong Freudenthal property implies that for every π T x for all x ∈ E there is a band projection π with x+ = πx. Therefore, T (x+ ) = ^ T ∈ L∼ (E, F ). B (3) Theorem. Let E and F be vector lattices with F order complete. A set P of band projections in L∼ (E, F ) is generating if and only if for any T, S ∈ L∼ (E, F ) and x ∈ E + the following formulas hold: (T ∧ S)x = inf{pT x + p⊥ Sx : p ∈ P}, (T ∨ S)x = sup{pT x + p⊥ Sx : p ∈ P}. C We confine exposition to sufficiency. Let P be a generating set of projections. In virtue of 1.3.2 (10) and 1.3.2 (1) it is sufficient to prove inf{pT x+p⊥ Sx} = 0 provided that T ∧ S = 0. First observe that V x+ − sup{pV x : p ∈ P} = V x+ + inf{−pV x+ + pV x− : p ∈ P} = inf{p⊥ V x+ + pV x− : p ∈ P}.

96

Chapter 3

Thus, the identities inf{pV x+ + p⊥ V x− : p ∈ P} = 0 and V x+ = sup{pV x : p ∈ P} are equivalent for all V ∈ L+ (E, F ) and x ∈ E. Note next that pT x = p(T (2y) + T (x − 2y)) ≤ 2T y + pT (x − 2y)− , p⊥ Sx = p⊥ (S(2x − 2y) + S(2y − x)) ≤ 2S(x − y) + p⊥ S(2y − x)+ for p ∈ P and 0 ≤ y ≤ x, x ∈ E + . Adding these inequalities and passing to the infimum we deduce inf {pT x + p⊥ Sx} ≤ 2 inf {T y + S(x − y)} p

y

+ inf {p(T + S)(2y − x)− + p⊥ (T + S)(2y − x)+ } = 0 p

and the result follows. B 3.1.6. Observe the following useful corollary to 3.1.2 and 3.1.5 (2). (1) If E has the strong Freudenthal property then it is possible to calculate the supremum in 3.1.2 (1), 3.1.2 (6), and the infimum in 3.1.2 (2) over partitions of x ∈ E+ into the disjoint sum: x1 ⊥ x2 in 3.1.2 (1, 2) and xk ⊥ xl (k 6= l) in 3.1.2 (6). C The assertions concerning 3.1.2 (1, 2) follow immediately from 3.1.5 (2, 3). Hence, we may write |T |x := (T ∨ (−T ))x = sup{T x1 − T x2 : x = x1 + x2 , x1 ⊥ x2 , x1 , x2 ∈ E+ } ≤ sup{|T x1 | + |T x2 | : x = x1 + x2 , x1 ⊥ x2 , x1 , x2 ∈ E+ }. This proves the inequality ≤ in 3.1.2 (5), while the reverse inequality is obvious. B (2) Let E and F be the same as in Theorem 3.1.2. Then operators T + and S in L (E, F ) are disjoint if and only if for every e ∈ E+ and 0 < ε ∈ R there exist a partition of unity πα ⊂ P(F ) and a family (eα ) ⊂ E, 0 ≤ eα ≤ e, such that for all α the inequalities hold: πα T eα ≤ εT e,

πα S(e − eα ) ≤ εSe.

If, in addition, E has the strong Freudenthal property then we may assume that (eα ) ⊂ E(e), i.e., eα is a fragment of e for every α. C Suppose that (S ∧ T )e := inf{T e + S(e − x) : 0 ≤ x ≤ e} = 0

Positive Operators

97

for some e ∈ E+ . Put u := Se∧T e+π(Se+T e), where π is the band projection onto {Se ∧ T e}⊥ . As is seen, Se + T e ∈ {u}⊥⊥ , [Se]u ≤ Se and [T e]u ≤ T e (as before, [a] stands for the band projection onto {a}⊥⊥ ). According to 1.4.6 (4) and (1) there are a partition of unity πα ⊂ P(F ) and a family (eα ) ⊂ [0, e] ⊂ E such that πα (T eα +S(e−eα )) ≤ εu for all α. Thus, πα T eα ≤ ε[T e]u and πα S(e−eα ) ≤ ε[Se]u. The converse is obvious. B (3) Let E be a vector lattice and let P be a generating set of band projections in L∼ (E, F ). Then operators T and S in L+ (E, F ) are disjoint if and only if for every e ∈ E+ and 0 < ε ∈ R there exist a partition of unity πα ⊂ P(F ) and a family (pα ) ⊂ P, such that for all α the inequalities hold: πα pα T e ≤ εT e,

πα p⊥ α Se ≤ εSe.

3.1.7. We will conclude this section with the problem of extension of positive operators and dominated extension of linear operators. Kantorovich Theorem. Let X be a preordered vector space, let X0 be a massive subspace in X, and let F be a K-space. Then each positive operator T0 : X0 → F has a positive extension T : X → F . C First, let X = X0 ⊕ X1 , where X1 is a one-dimensional subspace, X1 := {αx1 : α ∈ R}. Since X0 is massive and T0 is positive, the set U := {T0 x0 : x0 ∈ X0 , x0 ≥ x1 } is nonempty and bounded below. Assign T x := {T0 x0 + αf : x = x0 + αx1 , x0 ∈ X0 , α ∈ R} where f := inf U . It is clear that T is a linear operator from X to F and T |X0 = T0 . So, only the positivity of T needs checking. If x = x0 + αx1 and x ≥ 0, then the case of α equal to 0 is trivial. If α > 0 then x1 ≥ −x0 /α. This implies that −T0 x0 /α ≤ f , i.e., T x ∈ F+ . In a similar way for α < 0 observe that x1 ≤ −x0 /α. Thus, f ≤ −T0 x0 /α and, finally, T x = T0 x0 + αf ∈ F+ . Now let S be the collection of linear operators S : dom(S) → F such that S extends T0 and S(dom(S)+ ) ⊂ F+ . Clearly, S is inductive in order by inclusion and so, by the Kuratowski–Zorn Lemma, S has a maximal element T . If x1 ∈ X \ dom(T ), apply the above proved result with X := dom(T ) ⊕ X1 , X0 := dom(T ), T0 := T , and X1 := Rx1 to obtain an extension of T . But this contradicts the maximality of T ; thus, T is a sought operator. B 3.1.8. Let X be an arbitrary real vector space. An operator p : X → E is called sublinear if p(x + y) ≤ p(x) + p(y) and p(λx) = λp(x) for all x, y ∈ X and 0 ≤ λ ∈ R. The collection of all linear operators from X into E dominated by p is called the support set of p and denoted by ∂p; symbolically, ∂p := {T ∈ L(X, E) : (∀x ∈ X) T x ≤ p(x)},

98

Chapter 3

where L(X, E) is the space of all linear operators from X to E. A member of ∂p is a supporting operator of p. The epigraph → /− >> /(p) of p is defined by → /− >> /(p) := {(x, e) ∈ X × E : e ≥ p(x)}. The epigraph of any sublinear operator p is a cone and defines in X ×E a preorder relation, so that (x, e) ≤ (x0 , e0 ) if and only if e0 − e ≥ p(x − x0 ). (1) Let U ∈ L(E, F ) and V ∈ L(X, F ). Then the operator B : (x, e) 7→ V e−U x is positive on the space X ×E endowed with the above-mentioned preorder if and only if V ≥ 0 and U ∈ ∂(V ◦ P ). C Suppose B ≥ 0. Since (e, 0) ≥ 0 for every e ∈ E+ , we conclude V e = B(0, e) ≥ 0. Thus, for (x, e) ≥ 0 we have V e − U x ≥ 0 and U x ≤ V e ≤ V p(x). The converse is similar. B Assume that X0 is a subspace of X and T0 : X0 → E is a linear operator such that T0 x ≤ p(x) for all x ∈ X0 . If for every such X, X0 , T0 and p, there exists an operator T ∈ ∂p that is an extension of T0 from X0 to the whole of X, then we say that E admits dominated extension of linear operators or have the dominated extension property. A complete characterization of ordered vector spaces admitting dominated extension of linear operators resides in the following two theorems. (2) Hahn–Banach–Kantorovich Theorem. Every order complete vector lattice has the dominated extension property. C The claim follows from the Kantorovich Theorem on using (1) with V = IF . B (3) Bonnice–Silvermann–To Theorem. Every ordered vector space admitting dominated extension of linear operators is order complete. 3.1.9. In the following corollaries E is a K-space and p is a sublinear operator acting from a vector space X into E. (1) Each sublinear operator is the upper envelope of its support set, i.e. the next representation holds p(x) = sup{T x : T ∈ ∂p} (x ∈ X). Moreover, for an arbitrary point x0 ∈ X there exists a linear operator T from X into E supporting p at x0 , i.e. such that T x0 = p(x0 ) and T ∈ ∂p. C Put X0 = {λx0 : λ ∈ R} and define the linear operator T0 : X0 → E by T (λx0 ) := λp(x0 ). For λ ≥ 0 we have T (λx0 ) = λp(x0 ) = p(λx0 ). If λ < 0 then T0 (λx0 ) = λp(x0 ) = −|λ|p(x0 ) ≤ p(−|λ|x0 ) = p(λx0 ). Thus T0 supports the restriction p  X0 . By the Hahn–Banach–Kantorovich Theorem there exists an extension T of the operator T0 to the whole space X dominated by p on X. This ensures that T ∈ ∂p and T x0 = T0 x0 = p(x0 ). B

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(2) Let Y be another vector space and let T be a linear operator from Y into X. Then ∂(p ◦ T ) = ∂p ◦ T. C Consider an arbitrary element S from ∂(p ◦ T ). Clearly, −p(T (−y)) ≤ Sy ≤ p(T y). Therefore, T y = 0 implies Sy = 0. This means that ker(T ) ⊂ ker(S), where ker(R) := R−1 (0) is the kernel of the operator R. Consequently, the equation X ◦ T = S is solvable for an unknown linear operator X : T (Y ) → E. By assumption, every solution U0 to this equation satisfies the inequality U0 x0 ≤ p(x0 ) for all x0 ∈ X0 := T (Y ). Therefore, according to the Hahn–Banach–Kantorovich Theorem, there exists an extension U ∈ L(X, E) of the operator U0 supporting the sublinear operator p. Thus U ∈ ∂p and U ◦ T = S, i.e. S ∈ ∂p ◦ T . The reverse inclusion is obvious. B It should be stressed that if T is an identical embedding of a subspace X0 into the space X, then the proposition exactly expresses the dominated extension property. In this connection Proposition 1.4.14 (2) is often referred to as the Hahn– Banach formula. (3) Let X0 be a massive sublattice of a vector lattice X and let S0 : X0 → E be a positive operator with S0 x0 ≤ T x0 (x0 ∈ X0 ). Then there exists a positive extension S : X → E of S0 such that S ≤ T . C Put p(x) := T (x+ ) (x ∈ X). Then ∂p = [0, T ] and S0 x0 ≤ S0 (x+ 0) ≤ + T (x+ ) = p(x ) for all x ∈ X . By the Hahn–Banach–Kantorovich Theorem there 0 0 0 0 is an extension S of S0 with S ∈ ∂p. B 3.2. Fragments of a Positive Operator The aim of the section is to obtain some basic formulas for calculating specific fragments of a positive operator. Simple fragments associated with order ideals in the domain of the operator prove to be very useful for this purpose. 3.2.1. An operator T : E → F is called order continuous (order σ-continuous) if T xα order converges to T x for each net (xα )α∈A (each sequence (xα )α∈N ) in E with order limit x. The set of all order continuous regular operators (order σ-continuous regular operators) with the vector and order structure induced from ∼ ∼ L∼ (E, F ) is denoted by L∼ n (E, F ) (Lnσ (E, F )). If F = R then we shall write En rather than L∼ n (E, R), see 1.5.2. A positive operator T ∈ L∼ (E, F ) is order continuous (order σ-continuous) if o and only if T xα → 0 for every decreasing net (sequence) (xα ) in E with inf α xα = 0. C The proof can be found in [388] (Lemmas VIII.3.1 and VIII.4.3). B An ideal G in E is said to be a σ-order-dense ideal if for any e ∈ E+ there is an increasing sequence (gn ) ⊂ G+ such that e = sup gn . An operator T ∈ L∼ (E, F ) is

100

Chapter 3

called singular if it vanishes on some order-dense ideal G ⊂ E. If T vanishes on a σorder-dense ideal, then U is called σ-singular. The sets of singular and σ-singular ∼ operators are denoted by L∼ s (E, F ) and Lsσ (E, F ), respectively. It is easily seen ∼ ∼ from 3.1.2 (5, 7) that Ls (E, F ) and Lsσ (E, F ) are order ideals in L∼ (E, F ). We say that a vector lattice E is rich in σ-order-dense ideals if for any x ∈ E+ there exists an element e ∈ E+ such that x ∈ {e}⊥⊥ and {e}⊥ + {e}⊥⊥ is a σorder-dense ideal in E. Each of the following types of vector lattices is rich in σ-order-dense ideals: (a) vector lattice with a weak order-unity; (b) vector lattice with the countable sup property; (c) vector lattice with the principal projection property. (d) vector lattice E in which for any x ∈ E there exists a countable antichain (en ), ek ⊥ ej (j 6= k), such that {en : n ∈ N}⊥ = E and x ∈ {e1 }⊥⊥ . 3.2.2. Theorem. Let E and F be vector lattices with E rich in σ-order-dense ideals and F order complete. An operator T ∈ L∼ (E, F ) is σ-order continuous if and only if it is disjoint to all σ-singular operators: ∼ ⊥ L∼ σ (E, F ) = Lsσ (E, F ) . ⊥ ∼ is true without any additional asC The inclusion L∼ nσ (E, F ) ⊂ Lsσ (E, F ) ∼ sumption on E. Indeed, if 0 ≤ U ∈ Lnσ (E, F ) and 0 ≤ V ∈ L∼ sσ (E, F ), then T := U ∧ V ≥ 0 is σ-continuous and vanishes on a σ-order-dense ideal G ⊂ E; thus, ∼ T = 0. The sets L∼ nσ (E, F ) and Lsσ (E, F ) are thus seen to be disjoint, since they are order ideals. ⊥ Now, suppose that an operator 0 < U ∈ L∼ nσ (E, F ) is not σ-order continuous. Take a sequence xn & 0, with f := inf n U xn > 0. By assumption there exists an element e ∈ E+ such that g1 ∈ {e}⊥⊥ and G = {e}⊥ + {e}⊥⊥ is a σ-order-dense ideal. It is no restriction to assume that x1 = e, since we may put e := e ∨ x1 if necessary. We will find a nonzero fragment of U vanishing on some σ-order-dense ideal. Since U (e) ≥ f > 0, we may choose a number 0 < ε < 1 and a nonzero band projection ρ in F such that ρf > ερU (e) > 0. Let en := (xn − εe)+ , πn := πen , and V := inf n ρ(πn U ), where πen is the band projection defined in 3.1.4 (5). Clearly en & 0, the sequence (πn ) is decreasing, and V is a fragment of U . Moreover, this fragment is nonzero, since

V (e1 ) = inf sup ρU (men ∧ e1 ) ≥ inf ρU (en ) ≥ ρ(f − εU (e)) > 0, n

m

n

Prove that V vanishes on the σ-order-dense ideal G generated by the increasing sequence gn = (εe − xn )+ and the order ideal {e}⊥ , in symbols: [ G = {e}⊥ + {E(gn )}.

Positive Operators

101

For x := gn we have V x ≤ πn U (gn ) = sup U (men ∧ gn ) = 0, m

since en and gn are disjoint. If 0 ≤ x ∈ {e}⊥ = {x}⊥ 1 , then V x ≤ π1 U (x) = 0. Thus V |G = 0. It remains to check that G is a σ-order-dense ideal. For an arbitrary x ∈ E+ we may find increasing sequences (e00n ) ⊂ {e}⊥ and (e0n ) ⊂ {e}⊥⊥ such that x = supn (e00n + e0n ). Moreover, e = supk gk /ε and e0n = supm me ∧ e0n (k, n, m ∈ N), whence x = sup (e00n + (mgk /ε) ∧ e0n ) = sup(e00n + (ngn /ε) ∧ e0n ). B n

m,n,k

3.2.3. Let E and F be vector lattices with F order complete. (1) Theorem. An operator T ∈ L∼ (E, F ) is order continuous if and only if it is disjoint from all singular operators: ∼ ⊥ L∼ n (E, F ) = Ls (E, F ) .

C The proof can be obtained as a simple modification of the reasoning of 3.2.2, on replacing the sequences xn and en by nets xα and eα = (xα − εxα0 ), α ≥ α0 , and taking into consideration that the ideal {xα0 }⊥⊥ + {xα0 }⊥ is always order-dense. B ∼ ∼ (2) The spaces L∼ n (E, F ) and Lnσ (E, F ) are bands in L (E, F ). C The statement concerning L∼ n (E, F ) is immediate from (1). To prove the ∼ second part we denote by L[e] (E, F ) the set of all regular operators U such that the ⊥⊥ restriction of U to {e}⊥⊥ lies in L∼ , F ). According to 3.1.3 (4) and 3.2.2 nσ ({e} ∼ L[e] (E, F ) is a band. Now, it remains to observe that L∼ nσ (E, F ) =

\

L∼ [e] (E, F ). B

e∈E+

^ be an order completion of E. Then every positive order (3) Let E ^ → F which continuous operator T : E → F has a unique positive extension T^ : E ^ is also order continuous. The correspondence T 7→ T extends by differences to ∼ ^ an isomorphism from L∼ n (E, F ) onto Ln (E, F ). ^ According to (1) it C Let T^ be an arbitrary positive extension of T to E. ∼ ^ F ). Note that if 0 ≤ S^ ∈ L∼ (E, ^ F ) then S := S| ^E suffices to prove that T^ ⊥ Ls (E, s is also singular, since G ∩ E is an order-dense ideal in E for every order-dense ideal ^ Thus, T ⊥ S. Using 3.1.2 (2), it is easy to observe that S^ ⊥ T^. B G ⊂ E.

102

Chapter 3

3.2.4. Let F be a filter in the lattice I (E) of order ideals in E. Then the set of F -singular operators LF (E, F ) = {T ∈ L∼ (E, F ) : NT ∈ F } is an order ideal in L∼ (E, F ). (1) The operator πF = inf{πG : G ∈ F } is the band projection onto LF (E, F )⊥ . C By definition πF is a band projection in L∼ (E, F ). Denote by π the projection onto LF (E, F )⊥ . If T ∈ LF (E, F ) and G := NT , then πF T ≤ πG T = 0. Thus πF ≤ π. Conversely, suppose that T ∈ LF (E, F )⊥ and G ∈ F . Then T − πG T is contained in LF (E, F )⊥ . At the same time T − πG T ∈ LF (E, F ), since T − πG T vanishes on G. Thus, T − πG T ∈ LF (E, F ) ∩ LF (E, F )⊥ = 0. Thus, we conclude that T − πG T = 0, whence πF ≥ π. B Denote by Id (E) and Iσd (E) the sets of all order-dense ideals and σ-orderdense ideals in E. Assume that Tn and Tnσ stand for o-continuous and σ-order continuous parts of T . If we take in (1) F := Id (E) or F := Iσd (E) then, by virtue of 3.2.2 and 3.2.3, we arrive at the following: (2) For an order continuous part of a positive operator T we have Tn = inf{πG T : G ∈ Id (E)}. (3) If, in addition, a vector lattice E is rich in σ-order-dense ideals then the formula is also valid: Tnσ = inf{πG T : G ∈ Iσd (E)}. C According to 3.2.4 (1) the operators πn and πnσ , defined by πn = inf{πG : G ∈ Id (E)},

πnσ = inf{πG : G ∈ Iσd (E)},

⊥ ⊥ are the band projections onto L∼ and L∼ respectively. But we s (E, F ) sσ (E, F ) ∼ ∼ ⊥ know that Ln (E, F ) = Ls (E, F ) and if E is rich in σ-order-dense ideals then we ∼ ⊥ have L∼ σ (E, F )= Lsσ (E, F ) . Therefore πn T = Tn and πnσ T = Tnσ . B

Positive Operators

103

3.2.5. Using projections of type πG , we may localize assumptions in Theorem 3.2.2 and deduce further formulas for calculating σ-order continuous parts. ∼ Let L∼ sσ(G) (E, F ) be a subset of L (E, F ) consisting of the operators vanishing on σ-order-dense ideals in G, i.e., Lsσ(G) (E, F ) = {T : NT ∩ G ∈ Iσd (G)}. (1) Let G ∈ I (E) be an ideal rich in σ-order-dense ideals. For T ∈ L (E, F ) the operator πG T is σ-order continuous if and only if it is disjoint from all operators σ-singular on G: ∼

∼ ⊥ πG L ∼ nσ (E, F ) = πG Lsσ(G) (E, F ) . ⊥ ∼ C If G ∈ I (E) is rich in σ-order-dense ideals then L∼ nσ (G, F ) = Lsσ (G, F ) ext ⊥ by Theorem 3.2.2. Therefore, Lext nσ (G, F ) = Lsσ (G, F ) , where the disjoint comext plement is taken in L (G, F ). Indeed, the inclusion ⊂ is obvious. If 0 ≤ T ∈ ∼ Lext (G, F ), T ⊥ Lext sσ (G, F ), and an operator 0 ≤ S ≤ T is such that S ∈ Lsσ (G, F ), ext then actually S ∈ Lsσ (G, F ) and consequently S = 0. This proves the converse inclusion. Using 3.1.3 (2, 4), we may now deduce

ext ext ⊥ πG L∼ = πG Lsσ(G) (E, F )⊥ . B nσ (E, F ) = EG Lnσ (G, F ) = EG Lsσ (G, F )

For every 0 ≤ T ∈ L∼ (E, F ) the following representations hold: (2) πG Tnσ = inf{πD T : D ∈ Iσd (G)}; (3) Tnσ x = inf{πD T x : D ∈ Iσd (G)} (x ∈ G+ ); (4) Tnσ = supe∈E+ inf{πD T : D ∈ Iσd ({e}⊥⊥ )}. C By 3.2.4 (1) the operator π = inf{πD : D ∩ G ∈ Iσd (G)} is the band projection onto Lsσ(G) (E, F )⊥ . Therefore, πG π = inf{πG πD : D ∩ G ∈ Iσd (G)} = inf{πC : C ∈ Iσd (G)} is the band projection onto πG Lsσ(G) (E, F )⊥ . This proves (2). The formula (3) follows from 3.2.1 (a) and (4) is true, since {e}⊥⊥ is rich in σ-order-dense ideals for every e ∈ E+ . B 3.2.6. (1) For any operator 0 ≤ T ∈ L∼ (E, F ) the following representations hold: Tn x = inf{sup T xα : xα % x} (x ∈ E+ ), α

Tnσ x = inf{sup T xn : xn % x} (x ∈ E+ ). n

104

Chapter 3

C Denote by h(T, x) the right-hand side of the second formula. Choose a sequence xn % x and consider the set G (xn ) of all order ideals in {x}⊥⊥ containing (xn ). By definition supn T xn ≤ πG T x for each G ∈ G (xn ), whence h(T, x) ≤ inf{πG T x : G ∈ G (xn )}. At the same time each σ-order-dense ideal G contains a sequence x0n % x and, by definition, G ⊂ G (x0n ). This implies, in virtue of 3.2.4 (3), that h(T, x) ≤ Tnσ x, thereby 0 ≤ h(T − Tnσ , x) ≤ (T − Tnσ )nσ x = 0. Moreover, h(Tnσ , x) = Tnσ x, since Tnσ is sequentially order continuous. Using the additivity of h(·, x), we obtain h(T, x) = h(Tnσ , x) + h(T − Tnσ , x) = Tnσ x. B (2) If E is a vector lattice with the strong Freudenthal property then for any 0 ≤ T ∈ L∼ (E, F ) we have Tn x = inf{sup T πα x : πα % [x]} (x ∈ E+ ), α

Tnσ x = inf{sup T πn x : πn % [x]} (x ∈ E+ ), n

where πα , πn are band projections in E and [x] is the band projection onto {x}⊥⊥ . C The proof is in a similar spirit. B 3.2.7. Theorem. Let S be an upward-directed set in L∼ (E, F )+ and σ denote the band projection onto S ⊥⊥ . Then for every 0 ≤ T ∈ L∼ (E, F ) and e ∈ E+ the following representations hold: (σ ⊥ T )e = inf sup{πT x : π ∈ P(F ), 0 ≤ x ≤ e, πSx ≤ εSe}, 0<ε∈R S∈S

(σT )e = sup inf{πT x + π ⊥ T e : π ∈ P(F ), 0 ≤ x ≤ e, πS(e − x) ≤ εSe}. 0<ε∈R S∈S

C It is sufficient to prove only the second identity. Denote by g(T ) the righthand side of the sought relation. Take an arbitrary S ∈ S and U ∈ L∼ (E, F ) with U ≤ (nS) ∧ T . The last inequality implies in particular that U belongs to the order ideal generated by S . For a fixed 0 < ε ∈ R choose any x ∈ [0, e] and π ∈ P(F ) so that πS(e − x) ≤ εSe. Then U e ≤ πU (e − x) + πT x + π ⊥ T e ≤ εnSe + (πT x + π ⊥ T e). Thus, U e ≤ εnSe + inf{πT x + π ⊥ T e : π ∈ P(F ), x ∈ [0, e] πS(e − x) ≤ εSe ≤ εnSe + g(T ).

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105

Since ε > 0 is arbitrary, we conclude U e ≤ g(T ). By hypotheses σT ∈ S ⊥⊥ ; therefore, σT = sup Uα for an increasing net (Uα ) in the order ideal generated by S . Hence, σT e = sup{Uα e} ≤ g(T ) or σT e = sup Uα e ≤ g(T ). Prove the reverse inequality. To this end observe first that (σT )e + g(σ ⊥ T ) = sup

{(σT )e + πσ ⊥ T x + π ⊥ σ ⊥ T e}

inf

0<ε∈R π∈P(F ),x∈[0,e] S∈S πS(e−x)≤εSe

≥ sup

inf

{π((σT )x + σ ⊥ T x) + π ⊥ T e} = g(T ).

0<ε∈R π∈P(F ),x∈[0,e] S∈S πS(e−x)≤εSe

In addition, we may use the already-proven formula g(T ) ≥ σT e for T := σT ; therefore, (σT )e + g(σ ⊥ T ) ≥ g(T ) ≥ g(σT ) + g(σ ⊥ T ) ≥ (σT )e + g(σ ⊥ T ), (σT )e + g(σ ⊥ T ) = g(T ). Thus, it remains to prove that g(σ ⊥ T ) = 0. Let U := σ ⊥ T and S ∈ S . Since U ⊥ S, we may choose, according to 3.1.6 (2), a partition of unity πα ⊂ P(F ) and a family (eα ) ⊂ E, 0 ≤ eα ≤ e, such that εU e ≥ πα U eα = πα (πα U eα + πα⊥ U e) ≥ πα inf{πU x + π ⊥ U e : π ∈ P(F ), 0 ≤ x ≤ e, πS(e − x) ≤ εSe}. Hence, εU e ≥ πα g(U ) for all α and εU e ≥ g(U ). Since the inequality is valid for an arbitrary ε > 0, we deduce g(U ) = 0. B 3.2.8. We present several simple corollaries to Theorem 3.2.7. Take positive operators T and S in L∼ (E, F ). Denote by TS the band projection of T onto {S}⊥⊥ . (1) If E is an arbitrary vector lattice then the representations are valid: (T − TS )e = inf sup{πT x : π ∈ P(F ), 0 ≤ x ≤ e, πSx ≤ εSe}, 0<ε∈R

(TS )e = sup inf{πT x + π ⊥ T e : π ∈ P(F ), 0 ≤ x ≤ e, πS(e − x) ≤ εSe}. 0<ε∈R

C We need only to put S := {S}⊥⊥ in 3.2.7. B

106

Chapter 3 (2) The formulas from (1) admit the following refinement: (T − TS )e = [Se]⊥ T e + inf sup{πT x : 0<ε∈R

π ∈ P(F ), π ≤ [Se], 0 ≤ x ≤ e, πSx ≤ εSe},

(TS )e = [Se]T e + sup inf{πT x 0<ε∈R ⊥

⊥

+([Se] − π)T e : π ∈ P(F ), π ≤ [Se] , 0 ≤ x ≤ e, πS(e − x) ≤ εSe}. C These formulas are equivalent to [Se]⊥ TS e = 0 and [Se](T − TS )e = 0 respectively. Using (1), we may calculate [Se]⊥ TS e = sup inf {[Se]⊥ πT x + [Se]⊥ π ⊥ T e} 0<ε∈R x,π

≤ sup inf {[Se]⊥ T x + [Se]⊥ [Se]T e} 0<ε∈R x

≤ sup

inf {[Se]⊥ T x} = 0. B

0<ε∈R 0≤x≤e

(3) Let 0 ≤ S := ϕ ⊗ u be a rank-one operator, i.e., Sx = ϕ(x)u for some u ∈ F+ and ϕ ∈ L∼ (E, R)+ . Then TS e = sup

inf {[u]T x : ϕ(e − x) ≤ εϕ(e)}.

0<ε∈R 0≤x≤e

C If 0 6= π ≤ [u] then πSx ≤ εSe means that ϕ(x)πu ≤ εϕ(e)u and is equivalent to ϕ(x) ≤ εϕ(e). At the same time the inequality πSx ≤ εSe does not depend on π if ϕ(e) 6= 0, since in this case [Se] = [u]. Thus the supremum in (1) is attained at π = [u] and we come to the relation (T − TS )e = inf

sup {[u]T x : ϕ(x) ≤ εϕ(e)} + [u]⊥ T e}

0<ε∈R 0≤x≤e

which is equivalent to the sought representation. B (4) Now let J(E, F ) = (L∼ (E, R)⊗F )⊥⊥ be the band of almost integral operators. Then the band projection σ onto J(E, F ) has the following representation: (σT )e = sup inf{T x : ϕ(e) ≤ εϕ(e − x), 0 ≤ e ≤ x}. 0<ε∈R 0≤ϕ∈E ∼

Positive Operators

107

(5) If ϕ ∈ E ∼ is an essentially positive o-continuous functional then (T − σT )x = inf sup{T e : ϕ(e) ≤ εϕ(x), 0 ≤ e ≤ x}. ε∈R+

e

3.2.9. (1) Theorem. Let E be an arbitrary vector lattice and P be a generating set of projections. Then the representation is valid: (T − TS )e = inf sup{πpT e : π ∈ P(F ), p ∈ P, πpSe ≤ εSe}, 0<ε∈R

(TS )e = sup inf{(πp)⊥ T e : π ∈ P(F ), p ∈ P, πpSe ≤ εSe}. 0<ε∈R

C The proof is quite analogous to that of 3.2.8 (1). The only difference is that one should use 3.1.5 (3) instead of 3.1.2 (1, 2). B (2) Assume that E is a vector lattice with the strong Freudenthal property and 0 ≤ S, T ∈ L∼ (E, F ) are arbitrary. Then for each e ∈ E+ we have (T − TS )e = inf sup{πT σe : π ∈ P(F ), σ ∈ P(E), πSσe ≤ εSe}, 0<ε∈R

(TS )e = sup inf{(T − πT σ)e : π ∈ P(F ), σ ∈ P(E), πSσe ≤ εSe}. 0<ε∈R

C The proof is straightforward from 3.1.4 (1), 3.1.5 (2), and (1). B (3) If E = F and S = IE is the identity operator in E then from (1) we may easily deduce: (T − TS )e = sup{π ⊥ T πe : π ∈ P(E)}. In particular, T belongs to the band {IE }⊥⊥ if and only if it is band preserving, i.e. T is an orthomorphism. 3.2.10. Theorem. Consider the following assertions: (1) T ∈ {S}⊥⊥ ; (2) for every e ∈ E+ and 0 < ε ∈ R there exits 0 < δ ∈ F , [Se] ≤ [δ], δ ≤ Se, such that πSx ≤ δ implies πT x ≤ εT e for all π ∈ P(F ) and x ∈ [0, e]; (3) for every e ∈ E+ and 0 < ε ∈ R there exits 0 < δ ∈ F , [Se] ≤ [δ], δ ≤ Se, such that πSσe ≤ δ implies πT σe ≤ εT e for all π ∈ P(F ) and σ ∈ P(E); (4) for every e ∈ E+ and 0 < ε ∈ R there exits 0 < δ ∈ F , [Se] ≤ [δ], δ ≤ Se, such that πpSe ≤ δ implies πpT e ≤ εT e for all π ∈ P(F ) and p ∈ P. For an arbitrary vector lattice E we have (1) ⇔ (2). In case when E has the strong Freudenthal property (1) ⇔ (3). If P is a generating set of projections in L∼ (E, F ) then (1) ⇔ (4).

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C We shall prove only one of the required equivalences, say (1) ⇔ (2). Other equivalences are proven in a quite similar way. Fix e ∈ E+ and define the decreasing sequence (rn (e)) by putting   1 rn (e) := sup πT x : π ∈ P(F ), 0 ≤ x ≤ e, πSx ≤ Se . n According to 3.2.8 (1) T ∈ {S}⊥⊥ if and only if inf n rn (e) = 0 for all e ∈ E+ . But the last relation is equivalent to the following: for every e ∈ E+ and 0 < ε ∈ R there exits a partition of unity (πn )n∈N in P(F ) such that πSx ≤

1 Se ⇒ πn πT x ≤ εT e n

for whatever π ∈ P(F ) and x ∈ [0, e]. Put δ := Since the inequalities πn πSx ≤ assertion can be rewritten as

1 n Se

∞ X 1 πn Se. n n=1

(n ∈ N) and πSx ≤ δ are equivalent, the last

(∀e ∈ E+ )(∀ε ∈ R, ε > 0)(∃ δ ∈ F, δ > 0)(∀x ∈ [0, e])(∀π ∈ P(F )) (πSx ≤ δ ⇒ πT x ≤ εT e). This completes the proof. B 3.3. Orthomorphisms and Lattice Homomorphisms Several useful results on the structure of orthomorphisms and lattice homomorphisms in vector lattices as well as a Hahn–Banach–Kantorovich-type theorem for lattice homomorphisms comprise the subject of the current section. 3.3.1. Take vector lattices E and F . A linear operator T : E → F is called a lattice homomorphism if one (and hence all) of the following identities holds: T (x ∨ y) = T x ∨ T y

(x, y ∈ E),

T (x ∧ y) = T x ∧ T y

(x, y ∈ E),

T (|x|) = |T x| (x ∈ E). It is clear that a lattice homomorphism preserves suprema and infima of nonempty finite sets and also preserves the positive and negative parts of every element. An injective lattice homomorphism is called a lattice (rarely, an order) monomorphism,

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an isomorphic embedding and even lattice isomorphism of E into F . If a lattice homomorphism T : E → F is a bijection then we say that E and F are lattice (or order) isomorphic or that T provides the order isomorphism between E and F . The set of all lattice homomorphisms from E into F is denoted by Hom(E, F ). ∼ ∼ Let L∼ a (E, F ) be the band in L (E, F ) generated by Hom(E, F ) and Ld (E, F ) be its disjoint complement: ⊥⊥ L∼ , a (E, F ) = Hom(E, F )

⊥ L∼ d (E, F ) = Hom(E, F ) .

An operator T ∈ L∼ (E, F ) is said to be atomic (diffuse) if T ∈ L∼ a (E, F ) (respec∼ tively T ∈ Ld (E, F )). A linear operator T : E → F is said to be disjointness preserving if T x ⊥ T y whenever x ⊥ y. Meyer Theorem. Let T be an order-bounded disjointness preserving operator from a vector lattice E to a vector lattice F . Then T has the positive part T + , negative part T − , and modulus |T | of T ; moreover, T + x = (T x)+ and T − x = (T x)− for 0 ≤ x ∈ E, and |T x| = |T |(|x|) for x ∈ E. 3.3.2. Consider a vector lattice E and some its sublattice D ⊂ E. A linear operator T from D into E is said to be band preserving or nonextending if one (and hence all) of the following identities holds: T e ∈ {e}⊥⊥ e ⊥ f ⇒ Te ⊥ f T (K ∩ D) ⊂ K

(e ∈ D), (e ∈ D, f ∈ E), (K ∈ B(E)).

A band preserving operator need not be order-bounded (see Section 5.1). The set of all order-bounded band preserving operators from D to some vector sublattice D0 ⊂ E is denoted by Orth(D, D0 ). An order-bounded band preserving operator α : D → E defined on an order-dense ideal D ⊂ E is named an extended orthomorphism in E. Since an extended orthomorphism is disjointness preserving, it is also regular according to 3.3.1. Let Orth(D, E) denote the set of all extended orthomorphisms defined on a fixed order-dense ideal D. (1) Every extended orthomorphism in a vector lattice is order continuous. Any two extended orthomorphisms commute. The space of all extended orthomorphisms Orth∞ (E) is defined as follows. Denote by M the collection of all pairs (D, π), where D is an order-dense ideal in E and π ∈ Orth(D, E). Elements (D, π) and (D0 , π 0 ) in M are announced to be equivalent (in writing (D, π) ∼ (D0 , π 0 )) provided that the orthomorphisms π and π 0 coincide on the intersection D ∩D0 . The factor set of M/ ∼ modulo this equivalence relation is denoted by Orth∞ (E). The set Orth∞ (E) under the pointwise addition,

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scalar multiplication, and lattice operations becomes a vector lattice. Moreover, it becomes an ordered algebra under composition. An element α ∈ Orth∞ (E), defined on the whole space E, is called an orthomorphism. The collection of all orthomorphisms in E is denoted by Orth(E). Let Z (E) be the o-ideal generated by the identity operator IE in L∼ (E). The space Z (E) is often called the ideal center of the vector lattice E. Identify every orthomorphism π ∈ Orth(E) with the corresponding equivalence class in Orth∞ (E). (2) The space Orth∞ (E) is a laterally complete faithful f -algebra with unity IE . Moreover, Orth(E) is an f -subalgebra of Orth∞ (E) and Z (E) is an f -subalgebra of Orth(E). (3) Every Archimedean f -algebra E with unity 1 is algebraically and latticially isomorphic to the f -algebra of orthomorphisms in E. Moreover, the ideal in E generated by 1 is mapped onto Z (E). (4) If E is an order complete vector lattice then Orth∞ (E) is a universally complete vector lattice and Orth(E) and Z (E) are order-dense ideals. (5) Let E and F be order-dense ideals in a universally complete vector lattice G with a fixed order and ring unity. Then for every orthomorphism π ∈ Orth(E, F ) there exists a unique g ∈ G such that πx = g · x (x ∈ E). 3.3.3. Kutateladze Theorem. Let E be a vector lattice and let F be a Kspace. A positive operator T : E → F is a lattice homomorphism if and only if, for every operator S : E → F satisfying 0 6 S 6 T , there is an orthomorphism ρ ∈ Orth(F ) such that 0 6 ρ 6 IF and S = ρ ◦ T . In other words, T is a lattice homomorphism if and only if [0, T ] = [0, IF ] ◦ T . C Let T be a lattice homomorphism and 0 ≤ S ≤ T . Then ker(T ) = NT ⊂ NS ⊂ ker(S) and there exists a linear operator ρ0 : F0 → F , F0 := T (E), such that S = ρ0 ◦ T . It is easy to observe that F0 is sublattice and 0 ≤ ρ0 ≤ IF |F0 . In view of 3.1.9 (3) ρ0 admits extension to a linear operator ρ : F → F with 0 ≤ ρ ≤ IF and we are done. Conversely, assume that [0, T ] = [0, IF ] ◦ T and take a pair of disjoint elements x, y ∈ E+ . Since πx T ≤ T , it follows by hypothesis that there exists an orthomorphism ρ ∈ Orth(F ) with πx T = ρ ◦ T . In virtue of 3.1.3 (1) πx T vanishes on E(x)⊥ and agrees with T on E(x). Therefore, ρ coincides with the identity mapping on the ideal F (T x) and vanishes at T y. Thus, T x ∧ T y = ρ(T x ∧ T y) = ρT x ∧ ρT y = T x ∧ 0 = 0. This shows that T is a lattice homomorphism. B 3.3.4. Now we list some simple consequences of Theorem 3.3.3 on assuming that E and F are vector lattices, with F order complete.

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(1) Let T : E → F be a lattice homomorphism and F = T (E)⊥⊥ . Then there exists an isomorphism ϕ from the Boolean algebra G (T ) of fragments of T onto P (F ) such that ϕ(S) ◦ T = S for all S ∈ G (T ). C Assign to each projection π ∈ P (F ) the operator ψ (π) := π ◦ T . Then ψ is a Boolean homomorphism from P (F ) into E(T ). If ψ(π) = 0 for some π ∈ P (F ) then π vanishes on the image of T and π = 0, since F = T (E)⊥⊥ . This means that ψ indeed is a monomorphism and it remains to establish surjectivity of ψ. Let S ∈ G (T ). According to 3.3.3 S = αT for some α ∈ Orth(F ). Since αT ∧ (I − α)T = 0, we have 0 = inf{αT e1 + (I − α)T e2 : e1 + e2 = e, e1 , e2 ∈ E+ } ≥ (α ∧ (I − α))T e for every e ∈ E+ . Therefore, α ∧ (I − α) vanishes on T (E), whence α ∧ (I − α) = 0 and α ∈ P (F ). The surjectivity of ψ is thus proven. It remains to observe that ϕ := ψ −1 is the isomorphism sought since ϕ(S) ◦ T = ψ ◦ ϕ (S) = S. B Let ˆ : E → G be a lattice homomorphism. Assume that G = ˆ(E)⊥⊥ is a K-space. In the universal completion mG, we fix the multiplicative structure that is uniquely determined by a choice of unity. Let F be an order ideal in mG, and Lˆ (E, F ) be the set of all regular operators S : E → F such that S ∈ {ˆ}⊥⊥ , where S and ˆ are regarded as operators from E into mG. Assign F 0 := f ∈ mF : f · ˆ(E) ⊂ F . (2) The set F 0 is an order ideal in mG which is linearly and order isomorphic to the K-space L ˆ (E, F ). Such an isomorphism is performed by associating with an element f ∈ F 0 the operator Sf by the formula Sf (e) = f · ˆ(e)

(e ∈ E).

(4) Operators S1 and S2 of the band {T }⊥⊥ are disjoint if and only if their images im(S1 ) and im(S2 ) are disjoint. (5) Observe that the order ideal I (T ) and the band {T }⊥⊥ generated by T admit a simple description. Namely, if Orth(T, F ) := Orth(G, F ), where G is the order ideal in F generated by T (E) then for every lattice homomorphism T we have I (T ) = Z (F ) ◦ T, {T }⊥⊥ = Orth(T, F ) ◦ T. 3.3.5. We now discuss Hahn–Banach-type extension theorems for lattice homomorphisms. First we establish some auxiliary facts. Let X be an ordered vector space and let F be a K-space. A positive operator T : X → F is said to be discrete if [0, T ] = [0, IF ] ◦ T . Here [0, T ] := {S ∈ L∼ (X, F ) : 0 ≤ S ≤ T } and [0, IF ] := {α ∈ L∼ (F ) : 0 ≤ S ≤ α} are order intervals in L∼ (X, F ) and L∼ (F ), respectively. Thus, in virtue of 3.3.3, if E is a vector lattice then a discrete operator

112

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is just a lattice homomorphism. It is noteworthy that discrete operators exist not only on vector lattices. There is a natural interconnection between discrete operators and extreme points of support sets. Recall that the operator T ∈ ∂p is called extreme if the conditions T1 , T2 ∈ ∂p and T1 + T2 = 2T imply that T = T1 = T2 . The set of extreme operators in the support set ∂p is denoted by Ch(p). It is seen that this definition agrees with the conventional concept of an extreme point of a convex set. A positive cone X+ of an ordered vector space X is said to be reproducing if X = X+ − X+ . Let X be an ordered vector space and T be an F -valued discrete operator on X. Then either F = {0} or X+ is a reproducing cone. C Let X := X+ − X+ and x0 ∈ X \ X. Now let f be a functional on X such that ker(f ) ⊃ X and f (x0 ) = 1. Denote by f ⊗ e the operator x 7→ f (x)e (x ∈ X). Clearly, T + f ⊗ T x0 ∈ [0, T ] whence T x0 + T x0 = αT x0 for some α ∈ [0, IE ]. This implies T x0 = 0. Thus we conclude that either X = X or T = 0. In the latter case if F 6= {0} and e ∈ F \ {0} then T + f ⊗ e ∈ [0, T ], and again X = X. B 3.3.6. Kantorovich Theorem for a Discrete Operator. Let X be an ordered vector space and F be a K-space. Further, let X0 be a massive subspace of X and let T0 : X0 → F be a discrete operator on X0 . Then there exists a discrete extension T of the operator T0 to the space X. C First let X = X0 + Rx1 for some x1 ∈ X \ X0 . Define T : X → F by T x1 := inf{T0 x0 : x0 ∈ X0 , x1 ≤ x0 } and T0 := T |X0 . It is clear that T is a positive operator (since X0 is massive). If T 0 ∈ [0, T ], then by assumption there is an orthomorphism α ∈ [0, IF ] with T 0 |X0 = αT |X0 , i.e. T 0 x0 = αT x0 for all x0 ∈ X0 . Since the equalities T 0 x1 = inf{T 0 x0 : x0 ∈ X0 , x1 ≤ x0 } = αT x1 ; (T − T 0 )x1 = inf{(T − T 0 )x0 : x0 ∈ X0 , x1 ≤ x0 } = (IE − α) ◦ T x1 0 hold, it follows that S T = α ◦ T. Now let X = t Xt , where (Xt ) is an upward-directed (by inclusion) family of subspaces containing X0 . Assume the positive operators Tt : Xt → E to be discrete and Ts to be a restriction of Tt to Xs whenever Xs ⊂ Xt . Consider the extension T of the operator T0 to X defined by the relation T x := Tt x (x ∈ Xt ). Introduce two sublinear operators P, Pt : X → F by the formulas

P (x) := inf{T x0 : x0 ∈ X, 0 ≤ x0 , x ≤ x0 }, Pt (x) := inf{T x0 : x0 ∈ Xt , 0 ≤ x0 , x ≤ x0 }. According to (1), P and Pt are correctly defined and, in addition, ∂P = [0, T ],

∂Pt = [0, Tt ].

Positive Operators

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Since Tt is discrete, we have (Tt x)+ = Pt (x) for x and t with x ∈ Xt . From this we deduce (T x)+ = (Tt x)+ = Pt (x) ≥ P (x) ≥ (T x)+ . Consequently, (T x)+ = P (x) for all x ∈ X. Calculating support sets, we obtain [0, IE ] ◦ T = ∂(x 7→ (x)+ ) ◦ T = ∂(x 7→ (T x)+ ) = ∂P = [0, T ], i.e. T is a discrete operator. By construction, T |X0 = T0 . B 3.3.7. Let p be a sublinear operator from a vector space to a vector lattice E and let T be a positive operator from E to a K-space F . For an operator S in ∂P the following are equivalent: (1) the operator T ◦ S belongs to Ch(T ◦ P ); (2) for all operators 0 ≤ T1 , T2 ∈ L∼ (E, F ) and S1 , S2 ∈ L(X, F ) satisfying the conditions T1 + T2 = T,

T ◦ S = S1 + S2 ,

S1 ∈ ∂(T1 ◦ P ),

S2 ∈ ∂(T2 ◦ P ),

the equalities T1 ◦ S = S1 and T2 ◦ S = S2 hold; (3) for the operator A : (x, e) 7→ e − Sx, defined on the ordered space X × E with positive cone → /− >> /(P ), the equality holds [0, T ] ◦ A = [0, T ◦ A ]. C (1) ⇒ (2): If the operators T1 , T2 , S1 , S2 satisfy the premises of (2) then the relations hold 2T ◦ S = (T1 ◦ S + S2 ) + (S1 + T2 ◦ S); T1 ◦ S + S2 ∈ ∂(T ◦ P ), S1 + T2 ◦ S ∈ ∂(T ◦ P ). Thus by (1) we have T ◦ S = T1 ◦ S + S2 = S1 + T2 ◦ S, whence T1 ◦ S = S1 and T2 ◦ S = S2 . (2) ⇒ (3): First of all observe that the operator B : (x, y) 7→ U y − V x, where U ∈ L(E, F ) and V ∈ L(X, F ), is positive on the space X × E endowed with the above-mentioned order, if and only if U ∈ L+ (E, F ) and V ∈ ∂(U ◦P ), see 3.1.7 (1). It follows that the operator A is positive since S ∈ ∂P . Therefore [0, T ] ◦ A ⊂ [0, T ◦ A ]. If (2) holds for S, and the operator B is positive and less than the operator T ◦ A , then U ≥ 0 and T − U ≥ 0. Moreover, V ∈ ∂(U ◦ P ) and we have T ◦ S − V ∈ ∂((T − U ) ◦ P ). Applying (2) for the operators T1 := U , T2 := T − U , S1 := V , and S2 := T ◦S −V , we obtain V = S1 = T1 ◦S = U ◦S. Thus, B = U ◦A , where U ∈ [0, T ].

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(3) ⇒ (1): Let S satisfy (3) and T ◦ S = α1 S1 + α2 S2 , where S1 , S2 ∈ ∂(T ◦ P ) and α1 , α2 ≥ 0, α1 + α2 = 1. Consider an operator B(x, y) := α1 T y − α1 S1 x. Then B ∈ [0, T ◦ A ] according to the relations (T ◦ A − B)(·, 0) = −α2 S2 ,

(T ◦ A − B)(0, ·) = α1 T.

Thus the equation T1 ◦A = B holds for some T1 ∈ [0, T ]. In other words, T1 = α1 T and T1 ◦ S = α1 S1 . This means that S1 = T ◦ S. B 3.3.8. Kutateladze–Milman Theorem. Let P : Y → E be a sublinear operator acting from a vector space Y into a K-space E. For T ∈ L(X, Y ) the following holds: Ch(P ◦ T ) ⊂ Ch(P ) ◦ T. C Let U ∈ Ch(P ◦ T ). Clearly, U = V ◦ T for some V ∈ ∂P in view of 3.1.9 (2). Let V0 be the restriction of V onto im(T ). Obviously, V0 lies in Ch(P ◦ ι), where ι is the identical embedding of im(T ) into Y . Therefore, by 3.3.7, the operator V0 : im(T ) × E → E, acting as V0 : (y, e) 7→ e − V0 e (y ∈ im(T ), e ∈ E), is discrete in the ordered space Y × E with positive cone → /− >> /(P ). The subspace im(T )×E is obviously massive in Y ×E. Thus, according to 3.3.6, there is a discrete extension V of the operator V0 . Undoubtedly, the operator Sy := V (y, 0) belongs to the support set ∂P being an extreme point there, and moreover, agrees with V on the image im(T ). In other words, U = V ◦ T = S ◦ T and S ∈ Ch(P ). B 3.3.9. Let X be an arbitrary vector space. A sublinear operator p : X → F is called a seminorm if p(x) = p(−x) for all x ∈ X. Given a seminorm p, define the set of linear operators Z (p) ⊂ L(X, F ) and the mapping · : Z (p) → Orth(F ) as Z (p) := {T ∈ L(X, F ) : (∃γ ∈ Orth(F )) T ∈ ∂(γ ◦ p)}, T := inf{γ ∈ Orth(F )+ : T ∈ ∂(γ ◦ p)} (T ∈ Z (p)).
(1) The triple Z (p), · , Orth(F ) is a Banach–Kantorovich space provided that p(x) = 0 implies x = 0. C Obviously, the operator · takes positive values and satisfy 2.1.1 (2, 3). If T = 0 then for every 0 < ε ∈ R there exist a partition of unity (πξ )ξ∈„ in P(F ) and a family (ρξ )ξ∈„ in Orth(F ) such that πξ ρξ ≤ εIF and T ∈ ∂(ρξ p) for all ξ ∈ „. Thus, πξ |T x| ≤ πξ ρξ p(x) ≤ εp(x), whence |T x| ≤ εp(x) and T = 0.

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Take a band projection π and an orthomorphism γ in F and observe that π ◦ T ∈ ∂(γp) if and only if π ◦ T ∈ ∂(π ◦ γ ◦ p). Moreover, since T ∈ ∂(α ◦ p) for some α ∈ Orth(F ), for every γ ∈ Orth(F ) with π ◦ T ∈ ∂(γ ◦ p) there exists γ 0 := πγ + π ⊥ α such that T ∈ ∂(γ 0 ◦ p) and πγ = πγ 0 . Taking into consideration these properties we may deduce: π ◦ T = inf{0 ≤ γ ∈ Orth(F ) : π ◦ T ∈ ∂(γ ◦ p)} = inf{π ◦ γ ∈ Orth(F )+ , π ◦ T ∈ ∂(γ ◦ p)} = inf{π ◦ γ ∈ Orth(F )+ , T ∈ ∂(γ ◦ p)} = π ◦ T . Thus, identifying π with the band projection π in Orth(F ) given by ρ 7→ π ◦ ρ, we conclude that π T = π T and in particular · is d-decomposable. Suppose the sequences (Tn ) ⊂ Z (p), (λn ) ⊂ R, and an orthomorphism ρ ∈ Orth(F ) satisfy the relation limn λn = 0 and Tn − Tm ≤ λk ρ for all m, n ≥ k. Then |Tn x − Tm x| ≤ λk ρp(x) and the sequence (Tn x) is r-fundamental in F . Therefore, we may define the linear operator T x := r-limn Tn x (x ∈ X). Since T − Tn ≤ λk ρ for n ≥ k, we conclude that T ∈ Z (p) and r-lim T − Tn = 0. Consequently, Z (p) is br-complete and also decomposable according to 2.1.9 (2). Disjoint completeness of Z (p) can be easily checked. It remains to apply 2.2.3. B A sublinear operator p : E → F is said to be a submorphism if p(x ∨ y) = p(x) ∨ p(y) for all x, y ∈ E. If p(x) = p(|x|) x ∈ E and the mapping x 7→ p(x+ ) is submorphism then p is called an M -seminorm. We say that p is positive if p(x) ≥ 0 for every x ∈ X. Denote by ∂h p the set of all lattice homomorphisms in ∂p. In the next two propositions p is an arbitrary positive submorphism. (2) Let T be a lattice homomorphism in ∂p. Then there exists a ρ ∈ Orth(F ) such that 0 ≤ ρ ≤ IF and T ∈ Ch(ρ ◦ p). C Take a lattice homomorphism S in ∂p. Assign λ(S) := inf{γ ∈ Orth(F )+ : S ∈ ∂(γ ◦ p)}. According to (1) λ(αS) = αλ(S) for every 0 ≤ α ∈ Orth(F ). Let S := (−S, 1) : (x, f ) 7→ f − Sx. Suppose that an order relation in X × F is given by the cone → /− >> /(γp). The operator B := (−B, β) : (x, f ) 7→ βf − Bx belongs to [0, S ]. Then, by definition Bx ≤ β ◦ γp(x) and (S − B)x ≤ (1 − β)γp(x) for all x ∈ X. Putting x ≤ 0, we obtain B ∈ [0, B] and, in view of 3.3.3, B = ρS for some ρ ∈ [0, IF ]. Thus, we arrive at ρS ∈ ∂(γβp) and (1 − ρ)S ∈ ∂((1 − β)γp). From this, using the homogeneity of λ(·), we deduce ργ = λ(ρS) ≤ γβ and (1 − ρ)γ = λ((1−ρ)S) ≤ (1−β)γ, whence ργ = βγ and ρ = β. The equality [0, S ] = [0, IF ]◦S is thereby proved, since B = (−B, β) = (−ρS, ρ) = ρ(−S, 1) = ρS . According to 3.3.7 S ∈ Ch(γ ◦ p). B (3) Every extreme point of ∂p is a lattice homomorphism. C Given an operator 0 ≤ B ≤ S define B0 : X × {0} → F by B0 (x, 0) := −Bx. Observe that Bx ≤ B(x+ ) ≤ S(x+ ) ≤ p(x+ ) = p(x)+ = p(x), i.e., B ∈ ∂p and

116

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the same is true for S − B. From this it follows that 0 ≤ B0 ≤ S |X×{0} , where S : (x, f ) 7→ f − Sx and X × F is ordered by → /− >> /p. According to 3.1.9 (3) there exists an extension B = (−B, β) of B0 to X × F such that 0 ≤ B ≤ S . In view of 3.3.7 B = ρ ◦ S for some ρ ∈ [0, IF ], whence β = ρ and B = ρS. It remains to apply Theorem 3.3.3. B 3.3.10. Theorem. Let X and E be vector lattices, and let F be a K-space. Let p : E → F be a positive submorphism and T : X → E be a lattice homomorphism. Then the Hahn–Banach formula for lattice homomorphisms is valid: ∂h (p ◦ T ) = (∂h p) ◦ T. C It suffice to prove that ∂h (p ◦ T ) ⊂ (∂h p) ◦ T . To this end, take S ∈ ∂h (p ◦ T ) and observe that p ◦ T is a submorphism. From 3.3.9 it follows that there exist an orthomorphism ρ ∈ [0, IF ] and an operator U ∈ Ch(p ◦ T ) such that S = ρ ◦ U . By the Kutateladze–Milman Theorem we see that U = V ◦ T for some V ∈ Ch(p). Thus, S = ρ ◦ V ◦ T . By virtue of 3.3.9 ρ ◦ V is a lattice homomorphism. Since p ≥ 0, 0 ≤ ρ ≤ IF , and V ∈ ∂p, we have ρ ◦ V ∈ ∂p, which completes the proof. B 3.3.11. We give two Hahn–Banach-type theorems for lattice homomorphisms. (1) Theorem. Let p be a positive submorphism from E to F , and let T0 be a lattice homomorphism from a sublattice E0 ⊂ E to F such that T0 x ≤ p(x) (x ∈ E0 ). Then T0 extends to a lattice homomorphism T : E → F with T x ≤ p(x) (x ∈ E). C The claim follows immediately if in 3.3.10 we substitute the identical embedding of E0 into E for T . B (2) Theorem. Let E0 be a massive sublattice in E and let T0 be a lattice homomorphism from E0 to F . Then T0 extends to a lattice homomorphism T : E → F. C This fact follows directly from 3.3.3 and 3.3.8. It is possible to produce another proof combining 3.1.8 (3) and 3.3.10 and observing that the sublinear operator p from 3.1.8 (3) is a submorphism if T0 is a lattice homomorphism. B 3.3.12. For any positive operator T ∈ L∼ (E, F ) define the mapping pT : E → F+ by pT (x) := inf {T x1 ∨ · · · ∨ T xn : |x| ≤ x1 ∨ · · · ∨ xn ; x1 , . . . , xn ∈ E+ , n ∈ N} . (1) The mapping pT is an M -seminorm. (2) For every T ∈ L∼ (E, F ) the representation holds: pT (x) = sup{Sx : S ∈ Hom(E, F ), 0 ≤ S ≤ T } (x ∈ E+ ).

Positive Operators

117

C Given x0 ∈ E+ , denote by E0 := Rx0 the vector sublattice generated by x0 , and put T0 (λx0 ) := λpT (x0 ), λ ∈ R. By 3.3.11 (1) there exists T ∈ ∂pT ∩Hom(E, F ) such that T x0 = T0 x0 = pT (x0 ). B (2) If E possesses the principal projection property then  pT (x) = inf T x1 ∨ · · · ∨ T xn : |x| = x1 + · · · + xn ; x1 , . . . , xn ∈ E+ , xk ⊥ xl (k 6= l), n ∈ N . 3.3.13. (1) An order-bounded operator T : E → F is diffuse if and only if  0 = inf |T |x1 ∨ · · · ∨ |T |xn : |x| ≤ x1 ∨ · · · ∨ xn ; x1 , . . . , xn ∈ E+ , n ∈ N (x ∈ E). C It is easy to see that T ∈ L∼ d (E, F ) if and only if no nontrivial homomorphism is dominated by |T |. This is equivalent to pT = 0 by virtue of 3.3.12 (2). B onto

(2) Let T ∈ L∼ (E, F ) and assume that S is the band projection of T Then the following representation holds:

L∼ a (E, F ).

( Sx = sup

n X

k=1

pT (xk ) : x =

n X

) xk ; x1 , . . . , xn ∈ E+ , n ∈ N

k=1

for all x ∈ E+ . C Denote by U x the right-hand side of the desired formula. In view of 3.3.4 (5) each positive operator from L∼ a (E, F ) is the supremum of a disjoint set in Hom(E, F ), since there is a maximal disjoint family in Hom(E, F ). Therefore, S = sup{V ∈ Hom(E, F ) : 0 ≤ V ≤ T }. Since Pn each V on the right-hand side of this Pnformula satisfies the inequality V x ≤ k=1 pT (xk ) whenever xk ∈ E+ and x = k=1 xk , we find that Sx ≤ U x. At the same time 3.3.12 (2) implies pT (x) ≤ Sx, x ∈ E+ , so that S = sup{V ∈ L∼ (E, F )+ : pT (u) ≤ Su (u ∈ E+ )}. Each V on the right-hand side of the formula satisfies U x ≤ Sx. Thus, S = U , which completes the proof. B

Pn

k=1

pT (xk ) ≤ V x, whence

118

Chapter 3 3.4. Maharam Operators

Under discussion is some class of order continuous positive operators that behave in many instances like functionals. A Radon–Nikod´ ym-type theorem for such operators is established. 3.4.1. Let E be a vector lattice, F be some K-spaces, and let T be a positive operator from E into F . Say that T possesses the Maharam property if, for every x ∈ E+ and 0 ≤ f ≤ T x ∈ F+ there exists 0 ≤ e ≤ x such that f = T e. A positive order continuous operator with the Maharam property is called a Maharam operator. Observe that T ∈ L(E, F )+ possesses the Maharam property if only if the equality T ( [0, x] ) = [0, T x] holds for any x ∈ E+ . Thus, a Maharam operator is exactly an order continuous order intervals preserving positive operator. Recall that the carrier CT of T , is defined by CT := {x ∈ E : T (|x|) = 0}⊥ . Let in addition FT := {T (|x| ) : x ∈ E}⊥⊥ , and let Dm (T ) be the greatest order-dense ideal in the universal completion m(E) of the space E to which T is extendible by o-continuity. Thus, z ∈ Dm (T ) if and only if z ∈ m(E) and the set {T (x) : 0 ≤ x ≤ |z|} is bounded in F . Say that a linear operator S : E → F is absolutely continuous with respect to T and write S  T if S(x) ∈ {T (x)}⊥⊥ for all x ∈ E. For α ∈ Orth (m(F )) we set D (α) := {e ∈ F : αe ∈ F }. Observe that the orthomorphism algebra Orth (m(F )) is a universally complete K-space. Moreover, the correspondence α 7→ (α, D (α)) implements a bijection from Orth (m(F )) onto Orth∞ (F ). Thus, Orth∞ (F ) is an f -algebra and a universally complete K-space. 3.4.2. Examples. (1) Every positive functional possesses the Maharam property. (2) Each positive orthomorphism acting in a K-space is a Maharam operator. (3) Let F be an arbitrary K-space, and let A be an arbitrary set. Denote by l1 (A, F ) the set of all o-summable families of elements in F indexed by A: ( l1 (A, F ) :=

)

(eα )α∈A ∈ F A

X : o|eα | ∈ F

.

α∈A

Define an operators T from l1 (A, F ) into F by the formula T z :=

X

eα

(z := (eα )α∈A ∈ l1 (A, F ) ).

α∈A

Then l1 (A, F ) with its natural vector structure and order is a K-space and T is a Maharam operator.

Positive Operators

119

(4) Let (Q, †, µ) be a probability space, and let F be a Banach lattice. Consider the space E := L1 (Q, †, µ, F ) of Bochner integrable F -valued functions, and let T : E → F denote the Bochner integral Z f dµ (f ∈ E).

T f := Q

If the Banach lattice F is order complete and has order continuous norm then E is a K-space under the natural order f ≥ 0 ⇔ f (t) ≥ 0 for almost all t ∈ Q) and T is a Maharam operator. (5) Take again a probability space (Q, †, µ) and let †0 and µ0 be a σsubalgebra of † and the restriction of µ to †0 . The conditional expectation operator E (·, †0 ) is a Maharam operator from L1 (Q, †, µ) onto L1 (Q, †0 , µ0 ). The restriction of E (·, †0 ) to Lp (Q, †, µ) is also a Maharam operator from Lp (Q, †, µ) to Lp (Q, †0 , µ0 ). These facts follow immediately from the simple properties of conditional expectation. (6) Let E0 be a massive vector sublattice of a vector lattice E. The latter means that E0 + E+ = E. Given an operator T ∈ L∼ (E, F ) denote by %(T ) the restriction of T to E0 . If F is a K-space then the restriction operator % : L∼ (E, F ) → L∼ (E0 , F ) is a surjective and essentially positive Maharam operator (see 3.1.2 (3)). 3.4.3. Theorem. Let E and F be some K-spaces, and let T be a Maharam operator from E into F . Then there exists a lattice isomorphism h from the universally complete K-space Orth∞ (FT ) onto the order complete K-subspace in Orth∞ (ET ) such that the following conditions hold: (1) h (Pr (FT ) ) is an order complete subalgebra of the Boolean algebra Pr (ET ); (2) h(Z (FT ) ) is a sublattice and subring in Z (ET ); (3) for every positive o-continuous operator S : E → F absolutely continuous with respect to T , the equality π ◦ S(x) = S ◦ h(π) (x) holds for all π ∈ Orth∞ (FT )+ and x ∈ D (h(π)); in particular, S is a Maharam operator. C Without loss of generality, we may assume that E = ET and F = FT . Put e := T (|e|)

(e ∈ E).

It is easily seen that · is a vector norm on E. Axioms 2.1.1 (1–3) are obvious. If e = f1 + f2 for some e ∈ E+ and f1 , f2 ∈ F+ , f1 ⊥ f2 , then by the Maharam property there exist e1 , e2 ∈ [0, |e|] such that T ek = fk (k := 1, 2). It is easy to

120

Chapter 3

observe that T (e1 ∧ e2 ) = 0 and T (|e| − e1 + e2 ) = 0. Therefore, e1 ⊥ e2 and |e| = e1 + e2 , since T is essentially positive. So, we have proved that · is ddecomposable. It is seen from the definitions that the norm · is monotone and semicontinuous. Now, from 2.1.4 we deduce the existence of a Boolean isomorphism h : P(F ) → P(E) satisfying (1). The isomorphism h is uniquely extended to the isomorphism from the space Orth∞ (F ) onto the order complete subspace in Orth∞ (E) constituted by those elements in Orth∞ (E) whose spectral functions take their values in the Boolean algebra PnB = h (Pr (F ) ). Denote this isomorphism by the same symbol h. If α := l=1 αl , πl , where λ1 , . . . , λn ∈ R+ and {π1 , . . . , πn } is a partition of unity in the algebra Pr (F ), then, obviously, πl ◦ α ◦ S = πl ◦ S (λl h(πl ) ) = πl ◦ S ◦ h(α) for all l. Summing over l yields α ◦ S = S ◦ h(α). Finally, if α ∈ Orth∞ (F )+ then α = sup (αξ ) for some upward-directed family (αξ ) in Z (FP). Whereas the n elements of Z (F ) are the r-limits of orthomorphisms of the form l=1 λl πl . Thus, to complete the proof of (1) and (2), it remains to appeal to o-continuity of the operator S. To prove (3) consider a positive operator S : E → F absolutely continuous with respect to T . By the definition of the isomorphism (see 2.1.2 (4)) h, for π ∈ B (F ) and x ∈ E we have S ◦ h(π)x ∈ {T ◦ h(π)x}⊥⊥ ⊂ π (F ). Consequently, π ⊥ ◦ S ◦ h(π) = 0 or S ◦ h(π) = π ◦ S ◦ h(π). Replacing π by π ⊥ in the previous argument, we obtain π ◦ S ◦ h(π ⊥ ) = 0. Therefore, π ◦ S = π ◦ S ◦ h(π). But then π ◦ S = π ◦ S ◦ h(π). We thus arrive at the sought relation π ◦ S = S ◦ h(π). B 3.4.4. Let E and F be some K-spaces and let T : E → F be a regular operator such that |T | is a Maharam operator. If (T x)+ > 0 for some x ∈ E+ then there exists a projection π ∈ Pr (E) such that T (πx) > 0 and the operator T ◦ π is positive. C Suppose T x  0 and look at the set … of all projections π ∈ Pr (E) meeting the inequality 0 ≥ T ◦ πx. It is easy to see that … 6= ∅. Hence, by order continuity of the operator T , every chain in … is bounded above. Consequently, by the Kuratowski–Zorn Lemma, there exist a maximal element π0 of the set …. If the projection 0 < π1 ≤ π0⊥ is such that T ◦ π1 x ≤ 0 then T ◦ (π1 + π0 )x ≤ T ◦ π1 x + T ◦ π0 x ≤ 0, and we arrive at a contradiction: π0 < π0 + π1 ∈ …. Hence, T ◦ π1 x  0 for every 0 6= π1 ∈ [0, π0⊥ ]. We now demonstrate that every such projection satisfies the inequality T ◦ π1 x ≥ 0. To this end, we suppose that π1 6= 0, π1 ⊥ π0 , and

Positive Operators

121

(T ◦ π1 x)− > 0. Let ρ be the band projection onto the band generated by the element (T ◦ π1 x)− . Then 0 > ρ ◦ T ◦ π1 x and, in view of Theorem 3.4.3, we have T ◦ h(ρ)π1 x < 0. This in particular implies that h(ρ) ◦ π1 > 0; and since h(ρ) ◦ π1 ⊥ π0 , by the above-mentioned property of the projection π0 we obtain T ◦ h(ρ) ◦ π1 x  0. This contradiction shows that T ◦ π1 x > 0 for all π1 6= 0, π1 ⊥ π0 . Finally let [x] be the band projection onto the band generated by the element x. Then π := π0⊥ ◦ [x] is a sought projection. Indeed, T ◦ πx = T ◦ π0⊥ x and T x = T ◦ π0⊥ x − (−T ◦ π0 x). It follows that T ◦ π0⊥ x ≥ (T x)+ > 0. On the other hand, if 0 ≤ y ∈ {x}⊥⊥ and (eyλ )λ∈R is the spectral function (or characteristic) of the element y relative to x then eyλ = 0 for λ < 0 and, for λ ≥ 0, we have T ◦ π(eyλ ) = T ◦ π0⊥ (eyλ ) = T ◦ π0⊥ ◦ [eyλ ]x ≥ 0. Appealing to the Freudenthal Spectral Theorem, we finally obtain: ∞  ∞ Z Z y  T ◦ π(y) = T ◦ π λ deλ λ d(T ◦ π(eyλ ) ) ≥ 0. B 0

0

3.4.5. Theorem. Let E and F be some K-spaces and let T : E → F be a Maharam operator. Then there exists an isomorphism ϕ from the Boolean algebra E (T ) of fragments of T onto P (ET ) such that T ◦ ϕ(S) = S for all S ∈ E (T ). C Again we may assume that E = ET . Let T0 stand for the unique o-continuous extension of the operator T to Dm (T ). Then, for each operator S ∈ E (T0 ), its restriction to E is an isomorphism of the Boolean algebras E (T0 ) and E (T ). Similarly, the Boolean algebras Pr (E) and Pr (Dm (T ) ) are isomorphic. Thus, without loss of generality, we may assume E = Dm (T ). Assign to each projection π ∈ Pr (E) the operator ψ (π) := T ◦ π. Then ψ is an increasing mapping from Pr (T ) into {T }⊥⊥ and ψ(0) = 0 and ψ(IE ) = T . Clearly, if the projections π and ρ are disjoint then the carriers of the operators ψ(π) and ψ(ρ) are disjoint too; therefore, ψ(π) ⊥ ψ(ρ). Moreover, for π ∈ Pr (E), the equalities ψ(IE − π) = T ◦ (IE − π) = T − T ◦ π = T − ψ(π) are valid. Consequently, ψ(π ⊥ ) = ψ(π)⊥ . Thus, ψ(π) ∈ E (T ) for all π ∈ Pr (E). Consider now two arbitrary projections π1 and π2 ∈ Pr (E). Since the projections ρl := πl − π1 ◦ π2 (l := 1, 2) are disjoint, the operators ψ(ρ1 ) and ψ(ρ2 ) are disjoint too. On the other hand, ψ(π1 ) ∧ ψ(π2 ) − ψ(π1 ∧ π2 ) = T ◦ π1 ∧ T ◦ π2 − T ◦ π1 ◦ π2 = (T ◦ π1 − T ◦ π1 ◦ π2 ) ∧ (T ◦ π2 − T ◦ π1 ◦ π2 ) = ψ(ρ1 ) ∧ ψ(ρ2 ) = 0.

122

Chapter 3

Hence, ψ(π1 ∧ π2 ) = ψ(π1 ) ∧ ψ(π2 ). Thus, ψ is a homomorphism from the Boolean algebra Pr (E) into the Boolean algebra E (T ). Essential positivity of the operator T implies that if ψ(π) = 0 for some π ∈ Pr (E) then π = 0. This means that ψ indeed is a monomorphism and it remains to establish surjectivity for it. Let S ∈ E (T ) and look at the set … := {π ∈ Orth (E)+ : T ◦ π ≤ S}. Using the Kuratowski–Zorn Lemma, we now demonstrate that … contains a maximal element. Indeed,