Bands in partially ordered vector spaces with order unit

On the existence of continuous (approximate) roots of algebraic equations

Introduction.Let L be an Archimedean vector lattice.The space LZ of all normal integrals (order continuous linear functionals) on L was first studied by H. Nakano [6], and has been the object of recent investigation [l]- [5].The central theorem in this paper shows that if L is the cut-completion of L, then LZ and (L)Z are isomorphic vector lattices, a known result when L = C(X), the continuous functions on a compact Hausdorff space (cf.[3, 10]).This serves as a convenient tool for examining the role that completeness of L plays in determining properties of LZ It will be shown that a number of results previously obtained under the assumption that L is complete can be extended to Archimedean vector lattices.

Let X, Y be compact Hausdorff spaces and B£(X), B*(Y)f 0 < a, |3 < 12 (the first uncountable ordinal), the associated Banach spaces of bounded real-valued Baire functions of classes a and 0. If B*(X) =£ B*AX) (which is the case if a =£ 0 and X is not dispersed), then B*(X) is neither linearly isometric to BQ(Y) nor equivalent to B^(Y) in several other ways. B^(X) is linearly isometric to B%(Y) if and only if X is Baire isomorphic to Y. For 1 < a < ft the maximal ideal space of B£(X) for a nondispersed compact space X As not an F-space. 1. Let I bea compact (more generally, completely regular) Hausdorff space and C(X) the space of continuous real-valued functions on X. Let B0(X) = C(X)9 and inductively define Ba(X) for each ordinal a < Q, (Q denotes the first uncountable ordinal) to be the space of pointwise limits of sequences of functions in U | < a B^(X). Let B*(X) be the space of bounded functions contained in Ba(X). With the pointwise operations Ba(X) and B*(X) are lattice-ordered algebras. With the supremum norm B*(X) is a Banach algebra (see [4, §41]). The Baire sets of X of multiplicative class a, denoted by Za(X)9 are defined to be the zero sets of functions in B*(X). Those of additive class a, denoted by CZa(X), are defined as the complements of sets in Za(X). Finally, those of ambiguous class a, denoted by Aa(X), are the sets which are simultaneously in Za(X) and CZa(X). With the set-theoretic operations of union and intersection, Aa(X) is a Boolean algebra for each a < 12. The sets of exactly ambiguous class a, denoted by EAa(X are those in ^ a ( Z ) U ^ < a Ac(X). The sets of exactly additive and exactly multiplicative class a are defined analogously. The class of all Baire subsets of X is AMS (MOS) subject classifications (1970). Primary 06A65, 26A21, 28A05, 46E1S, 46E30, 46J10, 54C50, 54H05; Secondary 04A15.

If the vector space \({\mathcal{L}}^r(E, F)\) of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for \({\mathcal{L}}^r(E, F)\) to be a vector lattice. \({\mathcal{L}}^r(E, F)\) and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces \({\mathcal{V}}(E, F)\) of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space \({\mathcal{L}}^r(E, F)\) or, in the case that \({\mathcal{L}}^r(E, F)\) is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to study the finite elements in such vector lattices \({\mathcal{V}}(E, F)\), where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures of E, F and \({\mathcal{V}}(E, F)\) are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite elements from E′ and F, are finite elements in \({\mathcal{V}}(E,F)\). The paper contains also some generalization of results obtained for the case \({\mathcal{V}}(E, F) = {\mathcal{L}}^r(E, F)\) in [10].

1. Introduction. A linear operator T on a vector lattice is band preserving if x⊥ y implies Tx ⊥ y. Much is known about the order bounded band preserving operators on an Archimedean vector lattice. The collection of all of these forms an Abelian algebra under composition and a vector lattice for the operator order (see [7], [8] and [13] amongst others). Very little appears to be known about band preserving operators which are not order bounded apart from some isolated examples ([11], [13], [1] and [17]) and some non-existence results ([11] and [1]).

Function-space compactifications of function spaces

Let A" be a set, A a family of real-valued functions on X which contains the constants, pA the weak uniformity generated by A, and U{nAX) the collection of uniformly continuous functions to the real line R.The problem is how to construct U(\iAX) from A. The main result here is: For A a vector lattice, the collection of suprema of countable, finitely Aequiuniform, order-one subsets of A + is uniformly dense in U(pAX).Two less technical corollaries: If A is a vector lattice (resp., vector space), then the collection of functions which are finitely A-uniform and uniformly localfy-A (resp., uniformly locally piecewise-A) is uniformly dense in U(pAX).Further, for any A, a finitely A-uniform function is fust a composition F (a,,... ,af) for some a{.^ e A and F uniformly continuous on the range of (fli.ap) in Rf.Thus, such compositions are dense in U(iiAX).For BU(iiAX), the compositions with F e BU(Rr) are dense (B denoting bounded functions).So, in a sense, to know U(pAX) it suffices to know A and subspaces of the spaces Rp, and to know BU(pAX), A and the spaces Rr suffice.In case A is a vector lattice and A ** BA (i.e., A consists of bounded functions), the problem of describing U(pAX) in terms of A has an elegant solution: A is uniformly dense in U(pAX) (i.e., if / G U(pAX) and t > 0 there is a E A with \f(x) -a(x)\ < e for all x E A").This can be seen to be essentially equivalent to the usual Stone-Weierstrass Theorem.The result appears to have been first published in 1955 by Maak [M] and Nbeling and Bauer [NB]; we give a short proof in the course of proving our main theorem.If A contains unbounded functions, the situation is more complicated: A = U(R)\N is a "uniformly closed" vector lattice, while pA is (uniformly) discrete.With X = R, let A be the piecewise linear functions (finitely many pieces); pA is the usual uniformity on R and the closure of A consists only of continuous eventually linear functions.

We present some of the classical inequalities in analysis in the context of Archimedean (real or complex) vector lattices and f-algebras. In particular, we prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean vector lattice, from which a Cauchy–Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean f-algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove a Hölder inequality for weighted geometric mean closed Archimedean Φ-algebras, substantially improving results by K. Boulabiar and M. A. Toumi. As a consequence, a Minkowski inequality for weighted geometric mean closed Archimedean Φ-algebras is obtained.

Let X be a compact Hausdorff space and C(X) the Banach space of all real-valued continuous functions on X under the sup norm. If H is a linear subspace of C(X) which separates the points of X and contains the constant functions, then it is well known that there exists a smallest closed subset of X, called the Silov boundary of X relative to H and denoted by OHX, with the property that each hCH attains its maximum value on afHX. A point xeX is called an H-extremal point if the only positive linear functional u on C(X) such that u(h) = h(x) for all hCH is the evaluation functional 4. where O.,(f) =f(x) for all fE C(X). The set of H-extremal points of X is called the Choquet boundary of X relative to H and will be denoted by VHX. H. Bauer has shown [1, 2.1 ] that the Choquet boundary of X relative to H is nonempty and the Silov boundary of X relative to H is equal to the closure of the Choquet boundary. Bauer has also introduced (see [1 ]) an abstract Dirichlet problem for the above setting. If S:DHX is closed and xCX, then MS denotes the set of positive linear functionals u on C(S) such that u(h I S) =h(x) for all hCH. MS is always nonempty. The measures in MX'HX are called the H-harmonic measures belonging to x. A function f in C(X) is said to be H-harmonic if for every xEX and every u M we have u(f) =f(x). The set of H-harmonic functions is denoted by H. The Dirichlet problem is said to be solvable for X relative to H if 1^ j aHX = c(aHX). The Dirichlet problem is solvable if and only if to each x in X belongs exactly one H-harmonic measure (see [1, Satz 9]). Let X1, X2 be compact Hausdorff spaces and H1, H2 separating linear subspaces of C(X1), C(X2) respectively, which contain the constant functions. Equipped with the usual product topology, the cartesian product X1 X X2 is a compact Hausdorff space. In this paper we show that a subfamily H1+H2 of C(XIXX2) can be defined, in a natural way, for which VH1+H2Xl X X2 exists and VH1+H2Xl X X2 = VH1X1X VH2X2. We then give two theorems on the relation of the

Chapter 2 Positive operators

A general extended functional is a functional which is allowed to take on infinite values; in other words such functionals are similar to the Lebesgue integral on the space of all integrable functions. The problem of representing a general extended functional f on a vector lattice X as an operator with finite values only has been solved in [12]. In fact, this problem has been solved for a larger class of general extended operators on an ordered vector space. The solution of this problem was given by means of an extension of the range R of the functional f to some ultrapower of R. Notice, however, that it is not always the case that a functional f can be considered as a trace of some internal functional *f:*X→R. (We remark, without going into details, that such an internal functional exists exactly in the case when the results of the nonstandard analysis can be used for investigation of the given functional.) In [12] a ‘standard’ necessary and sufficient condition was given for solving this latter problem on the existence of *f. Namely, f is a trace of an internal *f if and only if it is finitely extendable. This result makes the finite extendability problem worthy of study. The present paper is a first attempt in this direction. Simultaneously we introduce and study a more general notion of weak finite extendability that coincides with finite extendability, for instance, for vector lattices with a strong unit.

Let X be a compact Hausdorff space, C(X) the class of all continuous functions f: X→R, considered as an ordered vector space over R with respect to its canonical order; furthermore let F be an order complete ordered vector space over R. A linear operator A: C(X)→F is called bounded, if it transforms each bounded set of C(X) into a bounded set of F. The purpose of this paper is to represent such a bounded linear operator A: C(X)→F as an integral Af=∫ dμ with respect to some content μ, defined on the algebraa(X) which is generated by the open sets U⊂X. This representation is unique, if μ is required to be regular (p. 11).

d-20 - H-Closed Spaces

Given two topologies J1, J2 on a set X, J1 is said to be coarser than J2, written J1 ≦ J2, if every set open under J1 is open under J2. A minimal Hausdorff space is then one for which there is no coarser Hausdorff topology etc. Vaidyanathaswamy [4] showed that every compact Hausdorff space is both maximal compact and minimal Hausdorff. This raised the question of whether there exist minimal Hausdorff non-compact spaces and/or maximal compact non-Hausdorff spaces. These questions were in fact answered in the affirmative by Ramanathan [2], Balachandran [1], and Hing Tong [3]. Their examples were, however, all on countable sets, and the topology constructed to answer one question bore no relation to the topology answering the second. In particular, the minimal Hausdorff non-compact topologies were not finer than any maximal compact topology.

By the Riesz Representation Theorem for locally compact Hausdorff spaces, for every positive linear functional I on K(X) there is a measure @m such that I(f)=@!fd@m, where K(X) is the set of continuous real functions with compact support on the locally compact Hausdorff space X. In this article we prove a uniformly computable version of this theorem for computably locally compact computable Hausdorff spaces X. We introduce a representation of the positive linear functionals I on K(X) and a representation of the Borel measures on X and prove that for every such functional I a measure @m can be computed and vice versa such that I(f)=@!fd@m.