Recent Trends on Order Bounded Disjointness Preserving Operators

Boulabiar, Karim

doi:10.33232/bims.0062.43.69

Cited by 16 publications

(2 citation statements)

References 38 publications

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“…Aspects of the theory of disjointness preserving operators are presented in [30,41,42]. The recent results on disjointness preserving operators are surveyed in [13]. In particular, the concept of disjointness preserving set of operators is discussed in the survey.…”

Section: Characterization and Representationmentioning

confidence: 99%

Boolean-Valued Analysis of Order-Bounded Operators

Kusraev¹,

Kutateladze²

2016

J Math Sci

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This is a survey of some recent applications of Boolean valued models of set theory to order bounded operators in vector lattices.Everywhere below B denotes a complete Boolean algebra, while V (B) stands for the corresponding Boolean valued universe (the universe of B-valued sets). A partition of unity in B is a family (b ξ ) ξ∈Ξ ⊂ B with ξ∈Ξ b ξ = 1 and b ξ ∧b η = 0 for ξ = η.By a vector lattice throughout the sequel we will mean a real Archimedean vector lattice, unless specified otherwise. We let := denote the assignment by definition, while N, Q, R, and C symbolize the naturals, the rationals, the reals, and the complexes. We denote the Boolean algebras of bands and band projections in a vector lattice X by B(X) and P(X); and we let X u stand for the universal completion of a vector lattice X.The ideal center Z (X) of a vector lattice X is an f -algebra. Let Orth(X) and Orth ∞ (X) stand for the f -algebras of orthomorphisms and extended orthomorphisms, respectively X. Then Z (X) ⊂ Orth(X) ⊂ Orth ∞ (X). The space of all order bounded linear operators from X to Y is denoted by L ∼ (X, Y ). The Riesz-Kantorovich Theorem tells us that if Y is a Dedekind complete vector lattice then so is L ∼ (X, Y ). Escher RulesNow, we present a remarkable interplay between V and V (B) which is based on the operations of canonical embedding, descent, and ascent.1.2.1. We start with the canonical embedding of the von Neumann universe into the Boolean valued universe. Given x ∈ V, we denote by x ∧ the standard name of x in V (B) ; i.e., the element defined by the following recursion schema:∅ ∧ := ∅, dom(x ∧ ) := {y ∧ : y ∈ x}, im(x ∧ ) := {1}.

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Section: Characterization and Representationmentioning

confidence: 99%

Boolean-Valued Analysis of Order-Bounded Operators

Kusraev¹,

Kutateladze²

2016

J Math Sci

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“…Especially the study of band preserving and disjointness preserving operators is an active field of research, see e.g. [1,2,3,11].…”

Section: Introductionmentioning

confidence: 99%

Bands in partially ordered vector spaces with order unit

2015

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In an Archimedean directed partially ordered vector space X , one can define the concept of a band in terms of disjointness. Bands can be studied by using a vector lattice cover Y of X . If X has an order unit, Y can be represented as a subspace of C( ), where is a compact Hausdorff space. We characterize bands in X , and their disjoint complements, in terms of subsets of . We also analyze two methods to extend bands in X to C( ) and show how the carriers of a band and its extensions are related. We use the results to show that in each n-dimensional partially ordered vector space with a closed generating cone, the number of bands is bounded by 1 4 2 2 n for n ≥ 2. We also construct examples of (n + 1)-dimensional partially ordered vector spaces with 2n n + 2 bands. This shows that there are n-dimensional partially ordered vector spaces that have more bands than an n-dimensional Archimedean vector lattice when n ≥ 4.

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Factorization of Order Bounded Disjointness Preserving Multilinear Operators

Kusraev

Kusraeva

2019

Springer Proceedings in Mathematics &Amp; Statistics

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Recent Trends on Order Bounded Disjointness Preserving Operators

Cited by 16 publications

References 38 publications

Boolean-Valued Analysis of Order-Bounded Operators

Boolean-Valued Analysis of Order-Bounded Operators

Bands in partially ordered vector spaces with order unit

Factorization of Order Bounded Disjointness Preserving Multilinear Operators

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