On the uniform density of C(X) ⊗ C(Y) in C(X × Y)

We prove that if X and Y are compact Hausdorff spaces, then every f ∈ C(X × Y)₊, i.e. f(x, y) ≥ 0 for all (x, y) ∈ X × Y, can be approximated uniformly from below and above by elements of the form , where f_i ∈ C(X)₊ and g_i ∈ C(Y)₊ for i = 1, 2, …, n. The proof uses only elementary topology. We use this result, in conjuction with Kakutani's M-spaces representation theorem, to obtain an alternative proof for a known property of Fremlin's Riesz space tensor product of Archimedean Riesz spaces.     

Disjointness in Partially Ordered Vector Spaces

A notion of disjointness in arbitrary partially ordered vector spaces is introduced by calling two elements x and y disjoint if the set of all upper bounds of x + y and −x − y equals the set of all upper bounds of x − y and −x + y. Several elementary properties are easily observed. The question whether the disjoint complement of a subset is a linear subspace appears to be more difficult. It is shown that in directed Archimedean spaces disjoint complements are always subspaces. The proof relies on theory on order dense embedding in vector lattices. In a non-Archimedean directed space even the disjoint complement of a singleton may fail to be a subspace. According notions of disjointness preserving operator, band, and band preserving operator are defined and some of their basic properties are studied.     

Vector lattice covers of ideals and bands in Pre-Riesz spaces

Abstract

Pre-Riesz spaces are ordered vector spaces which can be order densely embedded into vector lattices, their so-called vector lattice covers. Given a vector lattice cover Y for a pre-Riesz space X, we address the question how to find vector lattice covers for subspaces of X, such as ideals and bands. We provide conditions such that for a directed ideal I in X its smallest extension ideal in Y is a vector lattice cover. We show a criterion for bands in X and their extension bands in Y as well. Moreover, we state properties of ideals and bands in X which are generated by sets, and of their extensions in Y.     

Vector measures and strict topologies

Let X be a completely regular Hausdorff space and Cb(X) be the space of all real-valued bounded continuous functions on X, endowed with the strict topology βσ. We study topological properties of continuous and weakly compact operators from Cb(X) to a locally convex Hausdorff space in terms of their representing vector measures. In particular, Alexandrov representation type theorems are derived. Moreover, a Yosida-Hewitt type decomposition for weakly compact operators on Cb(X) is given.     

Weakly compact operators and strict topologies

Let X   be a completely regular Hausdorff space and _Cb(X)$C_b(X)$ be the Banach space of all real-valued bounded continuous functions on X, endowed with the uniform norm. It is shown that every weakly compact operator T   from _Cb(X)$C_b(X)$ to a quasicomplete locally convex Hausdorff space E   can be uniquely decomposed as T=_T1+_T2+_T3+_T4$T=T_1+T_2+T_3+T_4$, where _Tk:_Cb(X)→E$T_k:C_b(X)\to E$(k=1,2,3,4)$(k=1,2,3,4)$ are weakly compact operators, and _T1$T_1$ is tight, _T2$T_2$ is purely τ  -additive, _T3$T_3$ is purely σ  -additive and _T4$T_4$ is purely finitely additive. Moreover, we derive a generalized Yosida–Hewitt decomposition for E-valued strongly bounded regular Baire measures.     

COMPONENTS OF POSITIVE ELEMENTS IN RIESZ SPACES

Abstract

We find a necessary and sufficient condition on a Riesz space E such that a sub-Boolean algebra of components of a positive element generates the Boolean algebra of all components of the element. This condition yields a dual characterization of principal A-modules in the sense of D. Vuza. When applied to l-algebras, one finds improvements of results of J. Synnatzschke.     

Strict and Mackey topologies and tight vector measures

Let B() denote the Banach algebra of all bounded Borel measurable complex functions defined on a topological Hausdor? space X, and B_o() stand for the ideal of B() consisting of all functions vanishing at infinity. Then B() is a faithful Banach left B_o()-module and the strict topology β on B() induced by B_o() is a mixed topology. For a sequentially complete locally convex Hausdor? space (E, ξ), we study the relationship between vector measures m : → E and the corresponding continuous integration operators T_m : B() → E. It is shown that a measure m : → E is countably additive tight if and only if the corresponding integration operator T_m is (η, ξ)-continuous, where η denotes the infimum of the strict topology β and the Mackey topology τ (B(), ca()). If, in particular, E is a Banach space, it is shown that m is countably additive tight if and only if Tm(absconv(U ∪ W)) is relatively weakly compact in E for some τ (B(), ca())-neighborhood U of 0 and some β-neighborhood W of 0 in B(). As an application, we prove a Nikodym type convergence theorem for countably additive tight vector measures.     

Self complementary topologies and preorders

A topology on a set X is self complementary if there is a homeomorphic copy on the same set that is a complement in the lattice of topologies on X. The problem of characterizing finite self complementary topologies leads us to redefine the problem in terms of preorders (i.e. reflexive, transitive relations). A preorder P on a set X is self complementary if there is an isomorphic copy P of P on X that is arc disjoint to P (except for loops) and with the property that PP is strongly connected. We characterize here self complementary finite partial orders and self complementary finite equivalence relations.     

Pointfree prime representation of real Riesz maps

In classical topology it is proved, nonconstructively, that for a topological space X, every bounded Riesz map ϕ in C(X) is of the form for a point x ∈ X. In this paper our main objective is to give the pointfree version of this result. In fact, we constructively represent each real Riesz map on a compact frame M by prime elements. Received March 23, 2004; accepted in final form May 14, 2005.     

2-Universally complete Riesz spaces and inverse-closed Riesz spaces

In this paper we show mainly two results about uniformly closed Riesz subspaces of ?^X containing the constant functions. First, for such a Riesz subspace E, we solve the problem of determining the properties that a real continuous functiondefined on a proper open interval of ?should have in order that the conditions “E is closed under composition with ” and “E is closed under inversion in X” become equivalent. The second result, reformulated in the more general frame of the Archimedean Riesz spaces with weak order unit e, establishes that E (e-uniformly complete and e-semisimple) is closed under inversion in C(Spec E) if and only if E is 2-universally e-complete.     

Chain conditions and weak topologies

We study conditions on Banach spaces close to separability. We say that a topological space is pcc if every point-finite family of open subsets of the space is countable. For a Banach space E, we say that E is weakly pcc if E, equipped with the weak topology, is pcc, and we also consider a weaker property: we say that E is half-pcc if every point-finite family consisting of half-spaces of E is countable. We show that E is half-pcc if, and only if, every bounded linear map E→c₀(ω₁) has separable range. We exhibit a variety of mild conditions which imply separability of a half-pcc Banach space. For a Banach space C(K), we also consider the pcc-property of the topology of pointwise convergence, and we note that the space Cp(K) may be pcc even when C(K) fails to be weakly pcc. We note that this does not happen when K is scattered, and we provide the following example:

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There exists a non-metrizable scattered compact Hausdorff space K with C(K) weakly pcc.

     

The complexity of central series in nilpotent computable groups

The terms of the upper and lower central series of a nilpotent computable group have computably enumerable Turing degree. We show that the Turing degrees of these terms are independent even when restricted to groups which admit computable orders.     

On the epicomplete monoreflection of an archimedean lattice-ordered group

In a category C an object it G is epicomplete if the only epic monics out of G are isomorphisms, epic or monic meant in the categorical sense of right or left cancellable. For each of the categories Arch: archimedean ?-groups with ?-homomorphisms, and its companion category W: Arch-objects with distinguished weak unit and unit-preserving ?-homomorphisms, (and for the corresponding categories of vector lattices) epicompleteness has been characterized as divisible and conditionally and laterally σ-complete, and it has been shown to be monoreflective. Denote the reflecting functors by β and β ^W , respectively. What are they? For W the Yosida representation has been used to realize β ^W A as a certain quotient of B (Y A), the Baire functions on the Yosida space of A. For Arch, very little has been known. Here we give a general representation theorem, Theorem A, for β G as a certain subdirect product of W-epicomplete objects derived from G. That result, some W-theory, and the relation between epicity and relative uniform density are then employed to show Theorem B: β C _K (Y)=B _L (Y), where C _K (Y)is the ?-group of continuous functions on Y with compact support and B _L (Y) is the ?-group of Baire functions on Y having Lindelöf cozero sets.     

Strict topologies on measure spaces

Let X be a measurable space, let be a family of measurable subsets of it, and let be a subspace of complex measures on X that is also closed under restrictions of measures. In this paper we introduce the ‐convergence topology and the ‐strict topology on . Among other results, we find necessary and sufficient conditions for Hausdorff‐ness and coincide‐ness of these topologies. Applications to Lebesgue spaces, and also examples in Hausdorff topological spaces and locally compact groups are given.     

Morphic groups

A group G is called morphic if every endomorphism α:G→G for which Gα is normal in G satisfies G/Gα≅ker(α). This concept originated in a 1976 paper of Gertrude Ehrlich characterizing when the endomorphism ring of a module is unit regular. The concept has been extensively studied in module and ring theory, and this paper investigates the idea in the category of groups. After developing their basic properties, we characterize the morphic groups among the dihedral groups and the groups whose normal subgroups form a finite chain. We investigate when a direct product of morphic groups is again morphic, prove that a finite nilpotent group is morphic if and only if its Sylow subgroups are morphic, and present some results for the case where a p-group is morphic.     

The projection onto the center of an ℓ-algebra

In a uniformly complete -algebra with unite>0 the principal band {e} ^dd generated bye is a projection band. We prove that the formula of A. R. Schep for the projection onto this band which holds in the -algebra of order bounded operators on a Dedekind complete vector lattice, in general defines a projection onto the commutant of {e} ^dd . We present some examples to show that the commutant may be strictly larger than the band {e} ^dd , and also conditions which guarantee equality.This paper was written during the fall semester of 1993 while the first author visited the Department of Mathematics of the University of Leiden as a guest of Dr. C. B. Huijsmans, to whom the express his gratitude for the hospitality offered to him.     

On von Neumann regular f-algebras

It is shown that for a large class of f-algebras, von Neumann regularity and -lateral completeness are equivalent notions.     

Positive one-parameter semigroups on ordered banach spaces

In this review we describe the basic structure of positive continuous one-parameter semigroups acting on ordered Banach spaces. The review is in two parts.First we discuss the general structure of ordered Banach spaces and their ordered duals. We examine normality and generation properties of the cones of positive elements with particular emphasis on monotone properties of the norm. The special cases of Banach lattices, order-unit spaces, and base-norm spaces, are also examined.Second we develop the theory of positive strongly continuous semigroups on ordered Banach spaces, and positive weak^*-continuous semigroups on the dual spaces. Initially we derive analogues of the Feller-Miyadera-Phillips and Hille-Yosida theorems on generation of positive semigroups. Subsequently we analyse strict positivity, irreducibility, and spectral properties, in parallel with the Perron-Frobenius theory of positive matrices.     

Some fundamental properties for duals of Orlicz spaces

In this paper some basic properties of Orlicz spaces are extended to their dual spaces, and finally, criteria for extreme points in these spaces are given.