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We elaborate, strengthen, and generalize known representation theorems by different authors for regular operators on vector and Banach lattices. Our main result asserts, in particular, that every regular linear operator T acting from a vector lattice E with the principal projection property to a Dedekind complete vector lattice F, which is an ideal of some order continuous Banach lattice G, admits a unique representation T=Ta+Tc$T = T_a + T_c$, where Ta$T_a$ is the sum of an absolutely order summable family of disjointness preserving operators and Tc$T_c$ is an order narrow (= diffuse) operator. Our main contribution is waiver of the order continuity assumption on T. In proofs, we use new techniques that allow obtaining more general results for a wider class of orthogonally additive operators, which has somewhat different order structure than the linear subspace of linear operators. [ABSTRACT FROM AUTHOR]

Let U and V be two Archimedean Riesz spaces. An operator S : U → V is said to be unbounded order continuous (uo-continuous), if rα uo → 0 in U implies Srα uo → 0 in V . In this paper, we give some properties of the uo-continuous dual Uuo~ of U. We show that a nonzero linear functional f on U is uo-continuous if and only if f is a linear combination of finitely many order continuous lattice homomorphisms. The result allows us to characterize the uo-continuous dual Uuo~. In general, by giving an example that the uo-continuous dual U~ uo is not a band in U~, we obtain the conditions for the uo-continuous dual of a Banach lattice U to be a band in U~. Then, we examine the properties of uo-continuous operators. We show that S is an order continuous operator if and only if S is an unbounded order continuous operator when S is a lattice homomorphism between two Riesz spaces U and V . Finally, we proved that if an order bounded operator S : U → V between Archimedean Riesz space U and atomic Dedekind complete Riesz space V is uo-continuous, then /S/ is uo-continuous. [ABSTRACT FROM AUTHOR]

In this note we prove that positive multi-valued operators on Riesz spaces which are everywhere defined are always continuous. Further, we prove that the sum of two positive relations having the same null part is itself positive. This result allows us to study the domination problem in the class of positive relations. [ABSTRACT FROM AUTHOR]

In this paper, we will study on some topologies induced by order convergence in a Riesz space. We will investigate the relationships of them. [ABSTRACT FROM AUTHOR]

An agent faces a decision under uncertainty with the following structure. There is a set A of "acts"; each will yield an unknown real-valued payoff. Linear combinations of acts are feasible; thus, A is a vector space. But there is no pre-specified set of states of nature. Instead, there is a Boolean algebra I describing information the agent could acquire. For each element of I , she has a conditional preference order on A . I show that if these conditional preferences satisfy certain axioms, then there is a unique compact Hausdorff space S such that elements of A correspond to continuous real-valued functions on S , elements of I correspond to regular closed subsets of S , and the conditional preferences have a subjective expected utility (SEU) representation given by a Borel probability measure on S and a continuous utility function. I consider two settings; in one, A has a partial order making it a Riesz space or Banach lattice, and I is the Boolean algebra of bands in A . In the other, A has a multiplication operator making it a commutative Banach algebra, and I is the Boolean algebra of regular ideals in A . Finally, given two such vector spaces A 1 and A 2 with SEU representations on topological spaces S 1 and S 2 , I show that a preference-preserving homomorphism A 2 ⟶ A 1 corresponds to a probability-preserving continuous function S 1 ⟶ S 2 . I interpret this as a model of changing awareness. [ABSTRACT FROM AUTHOR]

In this paper we formulate and proof Girsanov's theorem in vector lattices. To reach this goal, we develop the theory of cross-variation processes, derive the cross-variation formula and the Kunita–Watanabe inequality. Also needed and derived are properties of exponential processes, Itô's rule for multi-dimensional processes and the integration by parts formula for martingales. After proving Girsanov's theorem for the one-dimensional case, we also discuss the multi-dimensional case. [ABSTRACT FROM AUTHOR]

In this paper we introduce two new classes of operators which are called strongly order continuous and order-to-unbounded order continuous operators and we study their lattice properties. With under some conditions, we show that order continuity is equivalent to strongly order continuity of operators on Riesz spaces. We show that the collection of all strongly continuous linear functionals on a Riesz space E is a band of E~. [ABSTRACT FROM AUTHOR]

This paper deals with a study on classes of non linear operators. Let SL(X,Y) be the set of all sublinear operators between two Riesz spaces X and Y. It is a convex cone of the space H(X,Y) of all positively homogeneous operators. In this paper we study some spaces generated by this cone, thereforewe study several properties, which are well known in the theory of Riesz spaces, like order continuity, order boundedness etc. Finally, we try to generalise the concept of adjoint operator. First, by using the analytic form of Hahn-Banach theorem, we adapt the notion of adjoint operator to the category of positively homogeneous operators. Then we apply it to the class of operators generated by the sublinear operators. [ABSTRACT FROM AUTHOR]

In this short paper we present a generalization of the Hahn-Banach extension theorem for A-linear operators. Some theoretical applications and results are given. [ABSTRACT FROM AUTHOR]

Let W, X, Y and Z be Dedekind complete Riesz spaces. For A ε Lr(Y,Z) and B ε Lr(W,X) let MA,B be the two-sided multiplication operator from Lr(X, Y ) into Lr(W, Z) defined by MA,B(T) = ATB. We show that for every 0 ⩽ A0 ε Lrn (Y,Z), ǀMA,B,Bǀ(T) = MA0,ǀBǀ(T) holds for all B ε Lr(W,X) and all T ε Lrn(X, Y ). Furthermore, if W, X, Y and Z are Dedekind complete Banach lattices such that X and Y have order continuous norms, then ǀMA,Bǀ = MǀAǀ, ǀBǀ for all A ε Lr(Y,Z) and all B ε Lr(W,X). Our results generalize the related results of Synnatzschke and Wickstead, respectively. [ABSTRACT FROM AUTHOR]

We consider regular martingales (i.e., martingales which can be written as differences of two positive martingales) in the measure-free setting of Riesz spaces. The aim of this note is to show that the space of such martingales is a Riesz space. We derive an analogue of a result of Troitsky's on regular norm bounded martingales in Banach lattices. [ABSTRACT FROM AUTHOR]

The concepts of conditional expectations, martingales and stopping times were extended to the Riesz space context by Kuo, Labuschagne and Watson (Discrete time stochastic processes on Riesz spaces, Indag. Math., 15 (2004), 435–451). Here we extend the definition of an asymptotic martingale (amart) to the Riesz spaces context, and prove that Riesz space amarts can be decomposed into the sum of a martingale and an adapted sequence convergent to zero. Consequently an amart convergence theorem is deduced. [ABSTRACT FROM AUTHOR]

Abstract: We show that a sequentially (τ)-complete topological vector lattice X τ is isomorphic to some L 1(μ), if and only if the positive cone can be written as X + = ℝ+ B for some convex, (τ)-bounded, and (τ)-closed set B ⊂ X + \ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y + = ℝ+ (for ‖u‖ >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials. [Copyright &y& Elsevier]

Abstract: We prove a representation theorem for Hausdorff locally convex (M)-lattices which are Dedekind σ-complete, and whose topologies are order σ-continuous and monotonically complete. These turn out to be the weighted spaces c 0(T, H), defined in the paper for T ≠ ∅ and H ⊂ ℝ T +. We also characterize the dual of c 0(T, H), as the space l 1 (T, H) defined in the last section. The known representation (on c 0(T)) of Banach (M)-lattices with order continuous norm follows as a particular case. We obtain these results by first proving a new general isomorphism theorem, which seems to be of independent interest. Our notion of “monotonic topological completeness” is weaker than the usual completeness and seems to be very convenient in the framework of topological ordered vector spaces. [Copyright &y& Elsevier]