scholarly journals Projection lateral bands and lateral retracts

2020 ◽  
Vol 12 (2) ◽  
pp. 333-339
Author(s):  
A. Kamińska ◽  
I. Krasikova ◽  
M. Popov
Keyword(s):  
Riesz Space ◽  
Projection Property ◽  
Lateral Band ◽  
Orthogonally Additive

A projection lateral band $G$ in a Riesz space $E$ is defined to be a lateral band which is the image of an orthogonally additive projection $Q: E \to E$ possessing the property that $Q(x)$ is a fragment of $x$ for all $x \in E$, called a lateral retraction of $E$ onto $G$ (which is then proved to be unique). We investigate properties of lateral retracts, that are, images of lateral retractions, and describe lateral retractions onto principal projection lateral bands (i.e. lateral bands generated by single elements) in a Riesz space with the principal projection property. Moreover, we prove that every lateral retract is a lateral band, and every lateral band in a Dedekind complete Riesz space is a projection lateral band.

scholarly journals On Sums of Strictly Narrow Operators Acting from a Riesz Space to a Banach Space

2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Oleksandr Maslyuchenko ◽  
Mikhail Popov
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Riesz Space ◽  
The Other ◽  
Finite Variation ◽  
Orthogonally Additive ◽  
Orthogonally Additive Operators

We prove that ifEis a Dedekind complete atomless Riesz space andXis a Banach space, then the sum of two laterally continuous orthogonally additive operators fromEtoX, one of which is strictly narrow and the other one is hereditarily strictly narrow with finite variation (in particular, has finite rank), is strictly narrow. Similar results were previously obtained for narrow operators by different authors; however, no theorem of the kind was known for strictly narrow operators.

The tensor product of Archimedean ordered vector spaces

1988 ◽  
Vol 104 (2) ◽  
pp. 331-345 ◽  
Author(s):  
J. J. Grobler ◽  
C. C. A. Labuschagne
Keyword(s):  
Tensor Product ◽  
Ordered Set ◽  
Riesz Space ◽  
Vector Spaces ◽  
Projection Property ◽  
Simple Method ◽  
Riesz Spaces ◽  
Space Tensor ◽  
Ordered Vector Spaces ◽  
Representation Techniques

A Riesz space tensor product of Archimedean Riesz spaces was introduced by D. H. Fremlin[2, 3]. His construction as well as a subsequent simplified version by H. H. Schaefer[10] depended on representation techniques and it is our aim to find a more direct way to prove the existence of the tensor product and to derive its properties. This tensor product proved to be extremely useful in the theory of positive operators on Banach lattices (see [3] and [10]) and should be considered as one of the basic constructions in the theory of Riesz spaces. It is therefore of interest to construct it in an intrinsic way. The problem to do this was already posed by Fremlin in [2]. In this paper we shall present two different approaches, the first of which is analogous to the formation of a free lattice generated by a given partially ordered set. (See [5], p. 41.) In the second one we first assume the Riesz spaces involved to have the principal projection property. In this case a simple method of construction by step-elements is available and the tensor product of arbitrary Archimedean Riesz spaces can then be obtained by embedding the spaces into their Dedekind completions. To complete the latter step we need results on the extension of Riesz bimorphisms which will be proved in §1. Both our approaches hinge on results about the tensor product of ordered vector spaces. It turns out that a unique tensor product for ordered vector spaces exists and is contained in the Riesz space tensor product. This is investigated in §2.

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 Measurable Riesz spaces

2021 ◽  
Vol 13 (1) ◽  
pp. 81-88
Author(s):  
I. Krasikova ◽  
M. Pliev ◽  
M. Popov
Keyword(s):  
Probability Measure ◽  
Boolean Algebra ◽  
Chain Condition ◽  
Riesz Space ◽  
Dense Subspace ◽  
Projection Property ◽  
Riesz Spaces ◽  
Countable Chain Condition ◽  
Countable Chain ◽  
Order Dense Subspace

We study measurable elements of a Riesz space $E$, i.e. elements $e \in E \setminus \{0\}$ for which the Boolean algebra $\mathfrak{F}_e$ of fragments of $e$ is measurable. In particular, we prove that the set $E_{\rm meas}$ of all measurable elements of a Riesz space $E$ with the principal projection property together with zero is a $\sigma$-ideal of $E$. Another result asserts that, for a Riesz space $E$ with the principal projection property the following assertions are equivalent. (1) The Boolean algebra $\mathcal{U}$ of bands of $E$ is measurable. (2) $E_{\rm meas} = E$ and $E$ satisfies the countable chain condition. (3) $E$ can be embedded as an order dense subspace of $L_0(\mu)$ for some probability measure $\mu$.

scholarly journals Atomic Operators in Vector Lattices

2020 ◽  
Vol 17 (5) ◽  
Author(s):  
Ralph Chill ◽  
Marat Pliev
Keyword(s):  
Nonlinear Operator ◽  
Vector Lattice ◽  
Projection Property ◽  
New Class ◽  
Vector Lattices ◽  
Complete Vector Lattice ◽  
Whole Space ◽  
Complete Vector ◽  
Orthogonally Additive ◽  
Orthogonally Additive Operators

Abstract In this paper, we introduce a new class of operators on vector lattices. We say that a linear or nonlinear operator T from a vector lattice E to a vector lattice F is atomic if there exists a Boolean homomorphism $$\Phi $$ Φ from the Boolean algebra $${\mathfrak {B}}(E)$$ B ( E ) of all order projections on E to $${\mathfrak {B}}(F)$$ B ( F ) such that $$T\pi =\Phi (\pi )T$$ T π = Φ ( π ) T for every order projection $$\pi \in {\mathfrak {B}}(E)$$ π ∈ B ( E ) . We show that the set of all atomic operators defined on a vector lattice E with the principal projection property and taking values in a Dedekind complete vector lattice F is a band in the vector lattice of all regular orthogonally additive operators from E to F. We give the formula for the order projection onto this band, and we obtain an analytic representation for atomic operators between spaces of measurable functions. Finally, we consider the procedure of the extension of an atomic map from a lateral ideal to the whole space.

scholarly journals Каждая латеральная полоса является ядром положительного ортогонально аддитивного оператора

2021 ◽  
pp. 115-118
Author(s):  
M.A. Pliev
Keyword(s):  
Open Problem ◽  
Vector Lattice ◽  
Linear Operators ◽  
Lateral Band ◽  
Vector Lattices ◽  
Additive Operator ◽  
Orthogonally Additive Operator ◽  
Disjointness Preserving ◽  
Orthogonally Additive ◽  
Orthogonally Additive Operators

{In this paper we continue a study of relationships between the lateral partial order $\sqsubseteq$ in a vector lattice (the relation $x \sqsubseteq y$ means that $x$ is a fragment of $y$) and the theory of orthogonally additive operators on vector lattices. It was shown in~\cite{pMPP} that the concepts of lateral ideal and lateral band play the same important role in the theory of orthogonally additive operators as ideals and bands play in the theory for linear operators in vector lattices. We show that, for a vector lattice $E$ and a lateral band $G$ of~$E$, there exists a vector lattice~$F$ and a positive, disjointness preserving orthogonally additive operator $T \colon E \to F$ such that ${\rm ker} \, T = G$. As a consequence, we partially resolve the following open problem suggested in \cite{pMPP}: Are there a vector lattice~$E$ and a lateral ideal in $E$ which is not equal to the kernel of any positive orthogonally additive operator $T\colon E\to F$ for any vector lattice $F$?

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