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.

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 Orthogonally Biadditive Operators

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Nonna Dzhusoeva ◽  
Ruslan Kulaev ◽  
Marat Pliev
Keyword(s):  
Vector Lattice ◽  
Cartesian Product ◽  
New Class ◽  
Mild Conditions ◽  
Vector Lattices ◽  
Complete Vector Lattice ◽  
Measurable Functions ◽  
Complete Vector ◽  
Orthogonally Additive ◽  
Order Continuous

In this article, we introduce and study a new class of operators defined on a Cartesian product of ideal spaces of measurable functions. We use the general approach of the theory of vector lattices. We say that an operator T : E × F ⟶ W defined on a Cartesian product of vector lattices E and F and taking values in a vector lattice W is orthogonally biadditive if all partial operators T y : E ⟶ W and T x : F ⟶ W are orthogonally additive. In the first part of the article, we prove that, under some mild conditions, a vector space of all regular orthogonally biadditive operators O B A r E , F ; W is a Dedekind complete vector lattice. We show that the set of all horizontally-to-order continuous regular orthogonally biadditive operators is a projection band in O B A r E , F ; W . In the last section of the paper, we investigate orthogonally biadditive operators on a Cartesian product of ideal spaces of measurable functions. We show that an integral Uryson operator which depends on two functional variables is orthogonally biadditive and obtain a criterion of the regularity of an orthogonally biadditive Uryson operator.

scholarly journals Minimal vector lattice covers

1971 ◽  
Vol 5 (3) ◽  
pp. 331-335 ◽  
Author(s):  
Roger D. Bleier
Keyword(s):  
Vector Lattice ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice

We show that each archimedean lattice-ordered group is contained in a unique (up to isomorphism) minimal archimedean vector lattice. This improves a result of Paul F. Conrad appearing previously in this Bulletin. Moreover, we show that this relationship between archimedean lattice-ordered groups and archimedean vector lattices is functorial.

Lifting components in clean abelian ℓ-groups

2018 ◽  
Vol 68 (2) ◽  
pp. 299-310
Author(s):  
Karim Boulabiar ◽  
Samir Smiti
Keyword(s):  
Vector Lattice ◽  
General Setting ◽  
Ring Theory ◽  
Natural Analogue ◽  
Order Unit ◽  
Step Functions ◽  
Strong Order

Abstract Let G be an abelian ℓ-group with a strong order unit u > 0. We call G u-clean after Hager, Kimber, and McGovern if every element of G can be written as a sum of a strong order unit of G and a u-component of G. We prove that G is u-clean if and only if u-components of G can be lifted modulo any ℓ-ideal of G. Moreover, we introduce a notion of u-suitable ℓ-groups (as a natural analogue of the corresponding notion in Ring Theory) and we prove that the ℓ-group G is u-clean when and only when it is u-suitable. Also, we show that if E is a vector lattice, then E is u-clean if and only if the space of all u-step functions of E is u-uniformly dense in E. As applications, we will generalize a result by Banaschewski on maximal ℓ-ideals of an archimedean bounded f-algebras to the non-archimedean case. We also extend a result by Miers on polynomially ideal C(X)-type algebras to the more general setting of bounded f-algebras.

Separation and Approximation in Topological Vector Lattices

1959 ◽  
Vol 11 ◽  
pp. 286-296 ◽  
Author(s):  
Solomon Leader
Keyword(s):  
Spectral Theory ◽  
Vector Lattice ◽  
Weierstrass Theorem ◽  
Fundamental Lemma ◽  
Vector Lattices ◽  
Integrable Functions ◽  
Nikodym Theorem

Spectral theory in its lattice-theoretic setting proves abstractly that the indicators of measurable sets generate the space L of Lebesgue-integrable functions on an interval. We are concerned here with abstractions suggested by the fact that indicators of intervals suffice to generate L. Our results show that the approximation of arbitrary elements of a topological vector lattice rests upon the ability to separate disjoint elements/ and g by an operation that behaves in the limit like a projection annihilating/ and leaving g invariant.The introduction of this concept of separation together with the notion of limit unit leads (via the Fundamental Lemma) to abstract generalizations of the Radon-Nikodym Theorem (Theorem 1) and the Stone-Weierstrass Theorem (Theorem 3).

Orthomorphisms of archimedean vector lattices

1981 ◽  
Vol 89 (1) ◽  
pp. 119-128 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Linear Operator ◽  
Vector Lattice ◽  
Jordan Decomposition ◽  
Duality Results ◽  
Vector Lattices ◽  
Elementary Methods ◽  
Disjointness Preserving ◽  
Order Boundedness

AbstractA linear operator T on a vector lattice L preserves disjointness if Tx ⊥ y whenever x ⊥ y. If such a T is positive it is automatically order bounded. An ortho-morphism is an order bounded disjointness preserving linear operator on L. In this note we show that the theory of orthomorphisms on archimedean vector lattices admits a totally elementary exposition. Elementary methods are also effective in duality considerations when the order dual separates points of L. For the Jordan decomposition T = T+ − T− with T+x = (Tx+)+ − (Tx−)+ we can dtrop the order boundedness assumption if we assume either that T preserves ideals or that L is normed and T is continuous. Alternatively we may keep order boundedness and assume only |Tx| ⊥ |Ty| whenever x ⊥ y. The main duality results show: T preserves ideals if and only if T** does; T is an orthomorphism if and only if T* is; T is central (|T| is bounded by a multiple of the identity) if and only if T* is central if and only if T and T* preserve ideals.

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.

Unital topology on a unital l-group

2020 ◽  
Vol 70 (5) ◽  
pp. 1189-1196
Author(s):  
Mahmood Pourgholamhossein ◽  
Mohammad Ali Ranjbar
Keyword(s):  
Vector Lattice ◽  
Riesz Space ◽  
Lattice Ordered Group ◽  
Order Unit ◽  
Ordered Group ◽  
Strong Link ◽  
Fundamental Properties ◽  
Essential Properties ◽  
Unit Norm

AbstractIn this paper we investigate some fundamental properties of unital topology on a lattice ordered group with order unit. We show that some essential properties of order unit norm on a vector lattice with order unit, are valid for unital l-groups. For instance we show that for an Archimedean Riesz space G with order unit u, the unital topology and the strong link topology are the same.

scholarly journals Topological Properties of the Complex Vector Lattice of Bounded Measurable Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Marian Nowak
Keyword(s):  
Banach Space ◽  
Vector Lattice ◽  
Topological Properties ◽  
Complex Vector ◽  
Mackey Topology ◽  
Measurable Functions ◽  
Complex Valued ◽  
Locally Convex

Let Σ be aσ-algebra of subsets of a nonempty set Ω. Let be the complex vector lattice of bounded Σ-measurable complex-valued functions on Ω and let be the Banach space of all bounded countably additive complex-valued measures on Ω. We study locally solid topologies on . In particular, it is shown that the Mackey topology is the finest locally convex-solidσ-Lebesgue topology on .

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