Weak Order Unit Research Articles

Strong completeness of a class of 𝐿²(𝑇)-type Riesz spaces

The L p , 1 ≤ p ≤ ∞ L^{p}, 1\le p\le \infty , spaces have been generalized to the setting of Riesz spaces as L p ( T ) {L}^{p}(T) spaces, on which there are R ( T ) R(T) -valued norms. The strong sequential completeness of the space L 1 ( T ) {L}^{1}(T) and the strong completeness of L ∞ ( T ) {L}^{\infty }(T) with resepct to their respective R ( T ) R(T) -valued norms were established by Kuo, Rodda, and Watson. In the current work, the T T -strong completeness of L 2 ( T ) {L}^{2}(T) is established via the Riesz–Fischer type theorem given by Kalauch, Kuo, and Watson. It is also shown that the conditional expectation operator T T is a weak order unit for the T T -strong dual.

Characterisation of conditional weak mixing via ergodicity of the tensor product in Riesz spaces

We link conditional weak mixing and ergodicity via the tensor product in Riesz spaces. In particular, we characterise conditional weak mixing of a conditional expectation preserving system by the ergodicity of its tensor product with itself or other ergodic systems. Along the way we characterise components of weak order units in the tensor product space in terms of tensor products of components of weak order units.

A Characterization of the Vector Lattice of Measurable Functions

Given a probability measure space (X,Sigma ,mu ), it is well known that the Riesz space L^0(mu ) of equivalence classes of measurable functions f: X rightarrow mathbf {R} is universally complete and the constant function varvec{1} is a weak order unit. Moreover, the linear functional L^infty (mu )rightarrow mathbf {R} defined by f mapsto int f,mathrm {d}mu is strictly positive and order continuous. Here we show, in particular, that the converse holds true, i.e., any universally complete Riesz space E with a weak order unit e>0 which admits a strictly positive order continuous linear functional on the principal ideal generated by e is lattice isomorphic onto L^0(mu ), for some probability measure space (X,Sigma ,mu ).

On the algebraic dimension of 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.

A Koopman-von Neumann type theorem on the convergence of Cesàro means in Riesz spaces

We extend the Koopman-von Neumann convergence condition on the Cesàro mean to the context of a Dedekind complete Riesz space with weak order unit. As a consequence, a characterisation of conditional weak mixing is given in the Riesz space setting. The results are applied to convergence in L 1 L^1 .

Ideal structure and factorization properties of the regular kernel operators

We consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on L^p-spaces. We show, in particular, that for any L^p(mu ) space, with mu sigma -finite and 1<p<infty , the norm-closure of the ideal of finite-rank operators on L^p(mu ), is the only non-trivial proper closed (order and algebra) ideal of this band. Key to our results in the L^p setting is the fact that every regular kernel operator on an L^p(mu ) space (mu and p as before) factors with regular factors through ell _p. We show that a similar but weaker factorization property, where ell _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.

Dedekind $$\sigma $$-complete $$\ell $$-groups and Riesz spaces as varieties

We prove that the category of Dedekind $\sigma$-complete Riesz spaces is an infinitary variety, and we provide an explicit equational axiomatization. In fact, we show that finitely many axioms suffice over the usual equational axiomatization of Riesz spaces. Our main result is that $\mathbb{R}$, regarded as a Dedekind $\sigma$-complete Riesz space, generates this category as a quasi-variety, and therefore as a variety. Analogous results are established for the categories of (i) Dedekind $\sigma$-complete Riesz spaces with a weak order unit, (ii) Dedekind $\sigma$-complete lattice-ordered groups, and (iii) Dedekind $\sigma$-complete lattice-ordered groups with a weak order unit.

Generalization of the theorems of Barndorff-Nielsen and Balakrishnan–Stepanov to Riesz spaces

In a Dedekind complete Riesz space, $E$, we show that if $(P_n)$ is a sequence of band projections in $E$ then $$\limsup\limits_{n\to \infty} P_n - \liminf\limits_{n\to \infty} P_n = \limsup\limits_{n\to \infty} P_n(I-P_{n+1}).$$ This identity is used to obtain conditional extensions in a Dedekind complete Riesz spaces with weak order unit and conditional expectation operator of the Barndorff-Nielsen and Balakrishnan-Stepanov generalizations of the First Borel-Cantelli Theorem.

The Unbounded Fuzzy Order Convergence in Fuzzy Riesz Spaces

The fuzzy order convergence in fuzzy Riesz spaces is defined only for fuzzy order bounded nets. The aim of this paper is to define and study unbounded fuzzy order convergence and some of its applications. Furthermore, some theoretical concepts like the fuzzy weak order unit and fuzzy ideals are studied in relation to unbounded fuzzy order convergence.

Averaging Operators and Continuous Projections on f-Algebras

Let A be an Archimedean f-algebra, let $$x\in A$$ , and let $$\pi _{x}:A\rightarrow A$$ be the linear map defined by $$\pi _{x}\left( y\right) =xy,$$ for all $$y\in A.$$ The aim of our paper is to give necessary and sufficient conditions concerning the averaging property of (r.u) continuous projections on Archimedean f-algebras, with a range, $$R\left( T\right) ,$$ a vector sublattice of A, that maps weak order units into weak order units. As an application, we prove that if A is an Archimedean f-algebra with a unit element e, T is a positive projection on A, with a range, $$R\left( T\right) ,$$ a vector sublattice of A, such that T(e) is a weak order unit of A, then T is an averaging operator if and only if $$R\left( T\right) $$ is $$\pi _{T\left( e\right) }$$ - invariant subspace. This improves considerably a result of Kuo et al. (J Math Anal Appl 303:509–521, 2005).

Maximal d-subgroups and ultrafilters

We study the space \({{\mathrm{Max~}}}_d(G)\) of maximal d-subgroups of a lattice-ordered group, paying specific attention to archimedean \(\ell \)-groups with weak order unit. For such an object (G, u), \({{\mathrm{Max~}}}_d(G)\) lays at a level in between the space of minimal prime subgroups and the Yosida space of (G, u). Theorem 5.10 gives the appropriate generalization of a quasi F-space to W-objects which avoids a discussion of o-complete \(\ell \)-groups.

Itô's rule and Lévy's theorem in vector lattices

The change of variable formula, or Itô's rule, is studied in a Dedekind complete vector lattice E with weak order unit E. Using the functional calculus we prove that for a Hölder continuous semimartingale Xt=Xa+Mt+Bt,t∈J, and a twice continuously differentiable function f, the formula(0.1)f(Xt)=f(Xa)+∫0tf′(Xs)dMs+∫0tf′(Xs)dBs+12∫0tf″(Xs)d〈M〉s,0≤s≤t∈J holds. The first integral in the formula is an Itô integral with reference to the local martingale M and the second and third integrals are Dobrakov-type integrals of a vector valued function with reference to a vector valued measure. Using the formula, we prove Lévy's characterization of Brownian motion as being a continuous martingale with compensator tE. The proof of this result yields a concrete description of abstract Brownian motion defined in vector lattices.

The Yosida space and representation of the projectable hull of an archimedean ℓ-group with weak unit

W is the category of archimedean ℓ-groups with designated weak order unit. The full subcategory of W of objects for which the unit is strong unit is denoted by W∗; such ℓ-groups are called bounded. Thus arises a coreflection . For G ∈W, Y G is the Yosida space, and G ≤ pG is the much-studied projectable hull. Recently, in [1], for G ∈ W∗, Y pG is identified as the Stone space of a certain boolean algebra of subsets of the minimal prime spectrum Min(G), and skepticism is expressed about extending this to W. Here, we show that indeed such an extension is possible, using a result from [5] and the following simple facts: in very concrete ways Min(G) and Min(BG) are homeomorphic spaces, and and are isomorphic boolean algebras; p and B commute.

Multiplicative structure of biorthomorphisms and embedding of orthomorphisms

Let X be an Archimedean vector lattice. A biorthomorphism on X is a bilinear map from X×X into X which is an orthomorphism on X in each variable separately. The set of such biorthomorphisms is denoted by Orth(X,X). We prove that if Orth(X,X) is not trivial then Orth(X,X) is equipped with a structure of f-algebra, giving thus a complete answer to a question asked quite recently by Buskes, Page, and Yilmaz. On the other hand, we assume that X is a semiprime f-algebra and we show that if X is either Dedekind-complete or uniformly-complete with a weak order unit, then the set of all orthomorphisms on X has an order ideal copy in Orth(X,X). Notice that the Dedekind-complete case has been obtained again by Buskes, Page, and Yilmaz in a completely different way.

Pointfree pointwise suprema in unital archimedean ℓ-groups

We generalize the concept of the pointwise supremum of real-valued functions to the pointfree setting. The concept itself admits a direct and intuitive formulation which makes no mention of points. But our aim here is to investigate pointwise suprema of subsets of RL, the family of continuous real valued functions on a locale, or pointfree space.Thus our setting is the category W of archimedean lattice-ordered groups (ℓ-groups) with designated weak order unit, with morphisms which preserve the group and lattice operations and take units to units. This is an appropriate context for this investigation because every W-object can be canonically represented as a subobject of some RL.We show that the suprema which are pointwise in the Madden representation can be characterized purely algebraically. They are precisely the suprema which are context-free, in the sense of being preserved by every W homomorphism out of G. We show that closure under such suprema characterizes the W-kernels among the convex ℓ-subgroups. Finally, we prove that all existing joins in a W-object G are pointwise iff its Madden frame L is boolean, and that all existing countable joins in G are pointwise if L is a P-frame, but not conversely.This leads up to the appropriate analog of the Nakano–Stone Theorem: a (completely regular) locale L has the feature that RL is conditionally pointwise complete (σ-complete), i.e., every bounded (countable) family from RL has a pointwise supremum in RL, iff L is boolean (a P-locale).We adopt a maximally broad definition of unconditional pointwise completeness (σ-completeness): a divisible W-object G is pointwise complete (σ-complete) if it contains a pointwise supremum for every subset which has a supremum in any extension. We show that the pointwise complete (σ-complete) W-objects are those of the form RL for L a boolean locale (P-locale). Finally, we show that a W-object G is pointwise σ-complete iff it is epicomplete.

Bounded Equivalence of Hull Classes in Archimedean Lattice-Ordered Groups with Unit

A hull class in a category is an object class H for which each object has a unique minimal essential extension in H. This paper addresses the enormity of the collection of hull classes in the category W of Archimedean l-groups with distinguished weak order unit through consideration of the action on the hull classes of the bounded coreflection \(\textbf {W} \overset {B}\rightarrow \textbf {W}^{\ast }\) onto the subcategory where the units are strong. It is shown that hull classes go forth under B and back under B−1, that the B-equivalence class of a hull class in W always has a top, and that these B-equivalence classes are frequently not sets. The property “top” is related to various other properties that hull classes might have. This paper is the third by us on the complex taxonomy of hull classes in W, and more are planned.

$$f$$ f -Representation of a function algebra

We investigate representations $$\Phi :A\longrightarrow \mathcal {L}_{b}(X)$$ , where $$A$$ is a unital function algebra and $$\mathcal {L}_{b}(X)$$ is the space of all order bounded operators on a vector lattice $$X$$ . Given an element $$x\in X$$ , the orbit space $$\Phi \left[ x\right] $$ generated by $$\Phi $$ at $$x$$ is the subspace $$\begin{aligned} \Phi \left[ x\right] =\left\{ \Phi (a)(x):a\in A\right\} . \end{aligned}$$ In this paper we make a detailed study of the orbite space $$\Phi \left[ x\right] , x\in X$$ . It turn out that they are vector lattices with a weak order unit. Moreover, It is proved that for any representation $$\Phi :A\longrightarrow \mathcal {L}_{b}(X)$$ can be extended to a representation of the order bidual $$A^{\sim }{}^{\sim }$$ of $$A$$ .

Convergence in Riesz spaces with conditional expectation operators

A conditional expectation, \(T\), on a Dedekind complete Riesz space with weak order unit is a positive order continuous projection which maps weak order units to weak order units and has \(R(T)\) a Dedekind complete Riesz subspace of \(E\). The concepts of strong convergence and convergence in probability are extended to this setting as \(T\)-strongly convergence and convergence in \(T\)-conditional probability. Critical to the relating of these types of convergence are the concepts of uniform integrability and norm boundedness, generalized as \(T\)-uniformity and \(T\)-boundedness. Here we show that if a net is \(T\)-uniform and convergent in \(T\)-conditional probability then it is \(T\)-strongly convergent, and if a net is \(T\)-strongly convergent then it is convergent in \(T\)-conditional probability. For sequences we have the equivalence that a sequence is \(T\)-uniform and convergent in \(T\)-conditional probability if and only if it is \(T\)-strongly convergent. These results are applied to Riesz space martingales and are applicable to stochastic processes having random variables with ill-defined or infinite expectation.

Clean unital ℓ-groups

Abstract A ring with identity is said to be clean if every element can be written as a sum of a unit and an idempotent. The study of clean rings has been at the forefront of ring theory over the past decade. The theory of partially-ordered groups has a nice and long history and since there are several ways of relating a ring to a (unital) partially-ordered group it became apparent that there ought to be a notion of a clean partially-ordered group. In this article we define a clean unital lattice-ordered group; we state and prove a theorem which characterizes clean unital ℓ-groups. We mention the relationship of clean unital ℓ-groups to algebraic K-theory. In the last section of the article we generalize the notion of clean to the non-unital context and investigate this concept within the framework of W-objects, that is, archimedean ℓ-groups with distinguished weak order unit.

The Kolmogorov–Čentsov theorem and Brownian motion in vector lattices

The well known Kolmogorov–Čentsov theorem is proved in a Dedekind complete vector lattice (Riesz space) with weak order unit on which a strictly positive conditional expectation is defined. It gives conditions that guarantee the Hölder-continuity of a stochastic process in the space. We discuss the notion of independence of projections and elements in the vector lattice and use this together with the Kolmogorov–Čentsov theorem to give an abstract definition of Brownian motion in a vector lattice. This definition captures the fact that the increments in a Brownian motion are normally distributed and that the paths are continuous.