Banach lattices of continuous sections Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2012, Том 14, Выпуск 4, С. 41-44

УДК 512.555+517.982

BANACH LATTICES OF CONTINUOUS SECTIONS1 A. G. Kusraev, S. N. Tabuev

The aim of this note is to outline some application of ample continuous Banach bundles to the theory of

Banach lattices.

Mathematics Subject Classification (2000): 06F25, 46A40.

Key words: Banach lattice, continuous Banach bundle, section, injective Banach lattice.

1. Introduction

The study of Banach lattices in terms of sections of continuous Banach bundles has been started by Giertz [1, 2]. Later Gutman create the theory of ample (or complete) continuous Banach bundles [3] and measurable Banach bundles admitting lifting [4]. A portion of the Gutman's theory was specified in the case of bundles of measurable Banach lattices by Ganiev [5] and Kusraev [6]. The aim of this short note is to outline some additional possibilities of applying ample Banach bundles to the theory of Banach lattices. Recall some definitions.

A bundle of Banach lattices over a set Q is a mapping X defined on Q and sending every point q £ Q to a Banach lattice X(q) := (X(q), || ■ ||q). Each space X(q) of a bundle X is called its stalk over q. A mapping u defined on a nonempty subset D С Q is called a section over D, if u(q) £ X(q) for every q £ D. A section over Q is called global. If Q is endowed with some topology we call sections over comeager subsets of Q almost global.

Let S(Q, X) stands for the set of all global sections of X, endowed with the structure of a vector lattice by letting u ^ v ^^ u(q) ^ v(q) (Vq £ Q), and (au+ev)(q) = au(q) + ev)(q) (q £ Q), where a, в £ R and u, v £ S(Q, X). For each section u £ S(Q, X) we define its point-wise norm by |||u||| : q ^ ||u(q)||x(q) (q £ Q). A set of sections U С S(Q, X) is called stalk-wise dense in X if the set {u(q) : u £ U} is dense in X(q) for every q £ Q.

2. Continuous bundles of Banach lattices

Let Q be a topological space and X be a bundle of Banach lattices over Q. A set of global sections C С S(Q, X) is called a continuity structure on X, if it satisfy the conditions:

(a) C is a vector lattice, i. e. ac1 + вс2 £ C, |c| £ C for all a, в £ R and c1; c2 £ C;

(b) the point-wise norm |||c||| : Q ^ R is continuous for every c £ C;

(c) C is stalk-wise dense in X.

If C is a continuity structure on X then the pair (X, C) is called a continuous bundle of Banach lattices over Q. More details see in [3] and [7]. Below (X, C) stands for a continuous bundle of Banach lattices over Q. We say that a section u £ S(D, X) over D С Q is C-continuous at the point q £ D if the function ||| u — c 11| is continuous at q for every c £ C .A section u £ S(D, X) is C-continuous if it is C-continuous at every q £ D.

© 2012 Kusraev A. G., Tabuev S. N.

1 The study was supported by The Ministry of education and science of Russian Federation, project 8210; by a grant from the Russian Foundation for Basic Research, project 12-01-00623-a.

Lemma. Let (X, C) be a continuous bundle of Banach lattices over Q. The set of all C-continuous sections over D c Q is a vector lattice.

< It is obvious that the set of all C-continuous sections is a vector space. Ensure that if a section u is C-continuous then so is |u| : q ^ |u(q)| (q G D). It is sufficient to prove that the function ||||u| — c||| : Q ^ R is continuous at an arbitrary q G D, for every c G C. Put A := || |u|(q) — c(q)||. We have to prove that, given q G D and e > 0, one can choose a neighborhood U of q such that A — e < || |u|(p) — c(p)|| < A + e for every p G U.

Select a section v G C satisfying ||u(q) — v(q)|| < e/2. Observe that || |u|(q) — |v|(q)|| ^ ||u(q) — v(q)|| < e/2. Taking into consideration the continuity of the function ||| |v| — c||| and the estimate || |v|(q) — c(q)|| ^ || |u|(q) — c(q)|| + || |u|(q) — |v|(q)|| < A + e/2 we can find a neighborhood Ui of q with || |v|(p) — c(p)|| < e/2 for all p G Ui.

Similarly, the estimate || |v|(q) — c(q)|| ^ |||u|(q) — c(q)|| — || |u|(q) — |v|(q)|| ^ A — e/2 implies that || |v|(p) — c(p)|| ^ A — e/2 (p G U2) for some neighborhood U2 of q. Now, for all p G U := U1 n U2 we can easily deduce

A — e = (A — e/2) — e/2 < || |v|(p) — c(p)|| — || |u|(p) — |v|(p)| < || |u|(p) — c(p)||, || |u|(p) — c(p)|| < || |v|(p) — c(p)|| + || |u|(p) — |v|(p)| < (A + e/2) + e/2 = A + e. >

3. Banach lattices of sections

Suppose that Q is a nonempty Stonean space (= extremally disconnected and compact Hausdorff space). Consider a continuous Banach bundle X over Q. If u is an almost global section of the bundle X then the function q ^ ||u(q)||q is defined and continuous on a comeager set dom(u) c Q. Consequently, there exists a unique function |u| G C^(Q) such that |u|(q) = ||u(q)||q (q G dom(u)).

In the set of almost global sections M(Q, X) we can define an equivalence relation by letting u ~ v if u(q) = v(q) whenever q G dom(u) n dom(v). Then equivalent u and v we have |u| = |v|; therefore, we may define |u| := |u|, where u is the coset of the almost global section u. Denote by C^(Q, X) the quotient space M(Q, X)/

In each coset u, there exists a unique section u G u such that dom(v) c dom(u) for all v G u. The section u is called extended. The space C^(Q, X) can be represented also as the space of all extended almost global sections of the bundle X, see [3]. The set C^(Q, X) can be naturally equipped with the structure of lattice-normed lattice. For instance, the element u+u is defined as the coset of the almost global section q ^ u(q)+v(q) (q G dom(u)ndom(v)). If E is an order ideal in C^(Q) then we assign E(X) := {u G C^(Q, X) : |u| G E}.

Recall that a Banach-Kantorovich space over a Dedekind complete vector lattice E is a vector space X with a decomposable norm |-| : X ^ E+ which is norm complete with respect to order convergence in E. Decomposability means that, given e1 ,e2 G E+ and x G X with |x| = e1 + e2, there exist x1 ,x2 G X such that x = x1 + x2 and |x&| = ek (k := 1,2). If a Banach-Kantorovich space is in addition a vector lattice with monotone norm then it's called a Banach-Kantorovich lattice. A Banach-Kantorovich lattice X can be endowed with a scalar norm x ^ |||-||| := ||H||e, whenever E is a Banach lattice. The following result see in

[3, 7].

Theorem 1. Let X be a continuous bundle of Banach lattices over a Stonean space Q. ThenC^(Q, X) is a Banach-Kantorovich latticeover C^(Q). If E is an order ideal in C^(Q) then (E(X), |-|) is a Banach-Kantorovich lattice over E. If, in addition, E is Banach lattice, then (E(X), |||-|||) is a Banach lattice.

Banach lattices of continuous sections

43

< We need only to put together the 'Banach part', given in [3] and [7, Theorem 2.4.7], and the above Lemma. >

Theorem 2. Every Banach-Kantorovich lattice X over an order dense ideal E c C^(Q) is isometrically lattice isomorphic to E(X) for some complete continuous bundle X of Banach lattices over Q. Moreover, such a bundle X is unique to within isometrically lattice isomorphism.

< The 'Banach part' follows again from [3] (see also [7, 2.4.10]). The rest is easily deduced on using the above Lemma. >

Let Q be the Stone space of the Boolean algebra B(fi) and t : fi ^ Q is the canonical immersion of fi into Q corresponding to a fixed lifting t of L^(fi). Let Y be a complete continuous bundle of Banach lattices over Q and X = Y ot. If C is a continuous structure in Y, then the set C o t is a measurability structure in X. The composite v o t is a measurable section of X for every v G C^(Q, Y), see [2, 1.2.7, 1.4.9, 2.5.8]. Let C(Q, X) stands for the set of all global continuous sections of X. The following result may be considered as a bridge between continuous and measurable bundles of Banach lattices.

Theorem 3. Let (fi, X, be a measurable space with the direct sum property. The mapping v ^ (v o t)~ is isometric lattice isomorphism of Banach-Kantorovich lattices C^>(Q, Y) and L0(fi, X), associated with the isomorphism (e ^ (e o t)~) : C^(Q) ^ L0(fi). The image of C(Q, Y) under this isomorphism is L^(fi, X).

< The 'Banach part' can be found in [4] (see also [2, 2.5.9]). The remaining is obvious. >

Remark 1. The theory of ample continuous bundles of Banach lattices is parallel to that of liftable Banach bundles presented in [6]. In particular, the results from [6, Theorems 2.9, 2.10, 3.3] have their counterparts for ample continuous bundles of Banach lattices.

4. Representation of injective Banach lattices

A real Banach lattice X is said to be injective if, for every Banach lattice Y, every closed vector sublattice Y0 c Y, and every positive linear operator T0 : Y0 ^ X there exists a positive linear extension T : Y ^ X with ||T01| = ||T||. This concept was introduced by Lotz [8]. Important contributions are due to Cartwright [9] and Haydon [10]. A new source of insight into the structure of injectives is a Boolean-valued approach, see [11, 12].

A band projection n in a Banach lattice X is said to be an M-projection if ||x|| = max{||nx||, x||} for all x G X, where := IX — n. The collection of all M-projections forms a subalgebra M(X) of the Boolean algebra of all band projections P(X) in X. The closed f-subalgebra in the center Z(X) generated by M(X) is denoted by Zm(X).

A positive operator T : X ^ F is said to have the Levi property if T(X= F and sup xa exists in X for every increasing net (xa) c X+, provided that the net (Txa) is order bounded in F. A Maharam operator is an order continuous order intervals preserving (= T([0,x]) = [0,Tx] for all x G X+) operator. An operator T : X ^ Y is called lattice B-isometry, if it is a lattice isometry and b o T = T o b for all b G B.

Theorem 4. If $ is a strictly positive Maharam operator with the Levi property taking values in a Dedekind complete AM-space A with unit and |||x||| = ||$(|x|)|^ (x G L1($)), then (L1 ($), |||-|||) is an injective Banach lattice with M(L1 ($)) = P(A). Conversely, any injective Banach lattice X is lattice B-isometric to (L1($), |||-|||) for some strictly positive Maharam operator $ with the Levi property taking values in a Dedekind complete AM-space A with unit, where B = M(L1($)) = P(A).

< See [12]; details can be found in [11]. >

Theorem 5. Every injective Banach lattice X with Л = Zm(X) = C(Q) and B := Р(Л) is lattice B-isometric to Л(Х) for some complete continuous bundle X of Banach lattices over Q such that all stalks X(q) (q G Q) are AL-spaces. Moreover, such a bundle X is unique to within isometrically lattice isomorphism.

< The proof consists of a combination of the representation Theorems 2 and 4. > Remark 2. This result was proved essentially by Gierz [1, 2] and Haydon [10]. The above

approach enables us to settle also the uniqueness problem.

References

1. Gierz G. Darstellung von Banachverbünden durch Schnitte in Bündeln // Mitt. Math. Sem. Univ. Giessen.-1977.-Vol. 125.

2. Gierz G. Bundles of Topological Vector Spaces and Their Duality.—Berlin etc.: Springer-Verlag, 1982.

3. Gutman A. E. Banach bundles in the theory of lattice-normed spaces. I. Continuous Banach bundles // Siberian Adv. Math.—1993.—Vol. 3, № 3.—P. 1-55.

4. Gutman A. E. Banach bundles in the theory of lattice-normed spaces. II. Measurable Banach bundles // Siberian Adv. Math.—1993.—Vol. 3, № 4.—P. 8-40.

5. Ganiev I. G. Measurable bundles of lattices and their applications // Studies on functional analysis and its applications.—Moscow: Nauka, 2006.—P. 9-49.

6. Kusraev A. G. Measurable bundles of Banach lattices // Positivity.—2010.—Vol. 14.—P. 785-799.

7. Kusraev A. G. Dominated Operators.—Dordrecht: Kluwer, 2000.—446 p.

8. Lotz H. P. Extensions and liftings of positive linear mappings on Banach lattices // Trans. Amer. Math. Soc.—1975.—Vol. 211.—P. 85-100.

9. Cartwright D. I. Extension of positive operators between Banach lattices // Memoirs Amer. Math. Soc.—1975.—Vol. 164.—P. 1-48.

10. Haydon R. Injective Banach lattices // Math. Z.—1977.—Vol. 156.—P. 19-47.

11. Kusraev A. G. Boolean Valued Analysis Approach to Injective Banach Lattices.—Vladikavkaz: Southern Math. Inst. VSC RAS, 2011.—28 p.—(Preprint № 1).

12. Kusraev A. G. Boolean-valued analysis and injective Banach lattices // Doklady Ross. Akad. Nauk.— 2012.—Vol. 444, № 2.—P. 143-145; Engl. transl.: Doklady Mathematics.—2012.—Vol. 85, № 3.—P. 341343.

Received Desember 9, 2012.

änatoly g. kusraev Southern Mathematical Institute Vladikavkaz Science Center of the RAS, Director Russia, 362027, Vladikavkaz, Markus street, 22 E-mail: kusraev@smath.ru

soslan n. tabuev

Southern Mathematical Institute

Vladikavkaz Science Center of the RAS, Researcher

Russia, 362027, Vladikavkaz, Markus street, 22

E-mail: soslan@tabuev.com

БАНАХОВЫ РЕШЕТКИ НЕПРЕРЫВНЫХ СЕЧЕНИЙ Кусраев А. Г., Табуев С. Н.

Заметка представляет собой набросок некоторых приложений просторных банаховых расслоений к теории банаховых решеток.

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