Essential completeness of archimedean ℓ-groups with weak order unit

2013 ◽  
Vol 217 (5) ◽  
pp. 915-926 ◽  
Author(s):  
B. Banaschewski ◽  
A.W. Hager
Keyword(s):  
Weak Order ◽  
Order Unit

scholarly journals Epicompletetion of archimedean l–groups and vector lattices with weak unit

1990 ◽  
Vol 48 (1) ◽  
pp. 25-56 ◽  
Author(s):  
Richard N. Ball ◽  
Anthony W. Hager
Keyword(s):  
Vector Lattice ◽  
Boolean Algebras ◽  
Weak Order ◽  
Minimal Extension ◽  
Order Unit ◽  
Weak Unit ◽  
Vector Lattices ◽  
Measurable Functions ◽  
Baire Functions

AbstractIn the category W of archimedean l–groups with distinguished weak order unit, with unitpreserving l–homorphism, let B be the class of W-objects of the form D(X), with X basically disconnected, or, what is the same thing (we show), the W-objects of the M/N, where M is a vector lattice of measurable functions and N is an abstract ideal of null functions. In earlier work, we have characterized the epimorphisms in W, and shown that an object G is epicomplete (that is, has no proper epic extension) if and only if G ∈ B. This describes the epicompletetions of a give G (that is, epicomplete objects epically containing G). First, we note that an epicompletion of G is just a “B-completion”, that is, a minimal extension of G by a B–object, that is, by a vector lattice of measurable functions modulo null functions. (C[0, 1] has 2c non-eqivalent such extensions.) Then (we show) the B–completions, or epicompletions, of G are exactly the quotients of the l–group B(Y(G)) of real-valued Baire functions on the Yosida space Y(G) of G, by σ-ideals I for which G embeds naturally in B(Y(G))/I. There is a smallest I, called N(G), and over the embedding G ≦ B(Y(G))/N(G) lifts any homorphism from G to a B–object. (The existence, though not the nature, of such a “reflective” epicompletion was first shown by Madden and Vermeer, using locales, then verified by us using properties of the class B.) There is a unique maximal (not maximum) such I, called M(Y(G)), and B(Y(G))/M(Y(G)) is the unique essentialBcompletion. There is an intermediate σ -ideal, called Z(Y(G)), and the embedding G ≦ B(y(G))/Z(Y(G)) is a σ-embedding, and functorial for σ -homomorphisms. The sistuation stands in strong analogy to the theory in Boolean algebras of free σ -algebras and σ -extensions, though there are crucial differences.

scholarly journals Ideal structure and factorization properties of the regular kernel operators

Positivity ◽  
2019 ◽  
Vol 24 (5) ◽  
pp. 1211-1229
Author(s):  
A. Blanco
Keyword(s):  
Banach Lattice ◽  
Weak Order ◽  
Ideal Structure ◽  
Order Unit ◽  
Factorization Property ◽  
Kernel Operator ◽  
Regular Kernel ◽  
And Algebra ◽  
Kernel Operators ◽  
The Ideal

AbstractWe consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on $$L^p$$ L p -spaces. We show, in particular, that for any $$L^p(\mu )$$ L p ( μ ) space, with $$\mu $$ μ $$\sigma $$ σ -finite and $$1<p<\infty $$ 1 < p < ∞ , the norm-closure of the ideal of finite-rank operators on $$L^p(\mu )$$ L p ( μ ) , is the only non-trivial proper closed (order and algebra) ideal of this band. Key to our results in the $$L^p$$ L p setting is the fact that every regular kernel operator on an $$L^p(\mu )$$ L p ( μ ) space ($$\mu $$ μ and p as before) factors with regular factors through $$\ell _p$$ ℓ p . We show that a similar but weaker factorization property, where $$\ell _p$$ ℓ p is replaced by some reflexive purely atomic Banach lattice, characterizes the regular kernel operators from a reflexive Banach lattice with weak order unit to a KB-space with weak order unit.

Multiplication in Vector Lattices

1968 ◽  
Vol 20 ◽  
pp. 1136-1149 ◽  
Author(s):  
Norman M. Rice
Keyword(s):  
Vector Lattice ◽  
Weak Order ◽  
Order Unit ◽  
Dedekind Complete Vector Lattice ◽  
Vector Lattices ◽  
Complete Vector Lattice ◽  
Complete Vector

B. Z. Vulih has shown (13) how an essentially unique intrinsic multiplication can be defined in a Dedekind complete vector lattice L having a weak order unit. Since this work is available only in Russian, a brief outline is given in § 2 (cf. also the review by E. Hewitt (4), and for details, consult (13) or (11)).

scholarly journals On the algebraic dimension of Riesz spaces

2021 ◽  
Vol 56 (1) ◽  
pp. 67-71
Author(s):  
N. M. Baziv ◽  
O. B. Hrybel
Keyword(s):  
Riesz Space ◽  
Weak Order ◽  
Order Unit ◽  
Projection Property ◽  
Linear Spaces ◽  
Infinite Dimensional ◽  
Algebraic Dimension ◽  
Riesz Spaces

We prove that the algebraic dimension of an infinite dimensional $C$-$\sigma$-complete Riesz space (in particular, of a Dedekind $\sigma$-complete and a laterally $\sigma$-complete Riesz space) with the principal projection property which either has a weak order unit or is not purely atomic, is at least continuum. A similar (incomparable to ours) result for complete metric linear spaces is well known.

scholarly journals Certain extensions and factorizations of α-complete homomorphisms in archimedean lattice-ordered groups

1997 ◽  
Vol 62 (2) ◽  
pp. 239-258 ◽  
Author(s):  
Anthony W. Hager ◽  
Ann Kizanis
Keyword(s):  
Full Subcategory ◽  
Cardinal Number ◽  
Weak Order ◽  
Infinite Cardinal ◽  
Order Unit ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Epireflective Subcategory ◽  
Archimedean Lattice

AbstractAs a consequence of general principles, we add to the array of ‘hulls’ in the category Arch (of archimedean ℓ-groups with ℓ-homomorphisms) and in its non-full subcategory W (whose objects have distinguished weak order unit, whose morphisms preserve the unit). The following discussion refers to either Arch or W. Let α be an infinite cardinal number or ∞, let Homα; denote the class of α-complete homomorphisms, and let R be a full epireflective subcategory with reflections denoted rG: G → rG. Then for each G, there is rαG ∈ Homα (G, R) such that for each ϕ ∈ Homα (G, R), there is unique with . Moreover if every rG is an essential embedding, then, for every α and every G, rαG = rG, and every Homα. If and R consists of all epicomplete objects, then every Homw1. For α = ∞, and for any R, every Hom∞.

scholarly journals Fréchet AL-spaces have the Dunford-Pettis property

1998 ◽  
Vol 58 (3) ◽  
pp. 383-386 ◽  
Author(s):  
J.C. Díaz ◽  
A. Fernández ◽  
F. Naranjo
Keyword(s):  
Banach Spaces ◽  
Positive Cone ◽  
Weak Order ◽  
Fréchet Spaces ◽  
Order Unit ◽  
Continuous Norm

A Fréchet lattice E is an AL-space if its topology can be defined by a family of lattice seminorms that are additive in the positive cone of E. Grothendieck proved that AL-Banach spaces have the Dunford-Pettis property. This result was recently extended by Fernández and Naranjo to AL-Fréchet spaces with a continuous norm and weak order unit. In this note we show how to remove both hypotheses.

Maximal d-Ideals in a Riesz Space

1983 ◽  
Vol 35 (6) ◽  
pp. 1010-1029 ◽  
Author(s):  
Charles B. Huijsmans ◽  
Ben de Pagter
Keyword(s):  
Riesz Space ◽  
Weak Order ◽  
Prime Ideals ◽  
Topological Properties ◽  
Order Unit ◽  
Structure Space ◽  
Maximal Ideals ◽  
Minimal Prime ◽  
The Ideal

We recall that the ideal I in an Archimedean Riesz space L is called a d-ideal whenever it follows from ƒ ∊ I that {ƒ}dd ⊂ I. Several authors (see [4], [5], [6], [12], [13], [15] and [18]) have considered the class of all d-ideals in L, but the set ℐd of all maximal d-ideals in L has not been studied in detail in the literature. In [12] and [13] the present authors paid some attention to certain aspects of the theory of maximal d-ideals, however neglecting the fact thatℐd, equipped with its hull-kernel topology, is a structure space of the underlying Riesz space L.The main purpose of the present paper is to investigate the topological properties of ℐd and to compare ℐd to other structure spaces of L, such as the space of minimal prime ideals and the space of all e-maximal ideals in L (where e > 0 is a weak order unit).

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