Convergence of Riesz space martingales

We prove a martingale convergence for sub and super martingales on Riesz spaces. As a consequence we can form Krickeberg and Riesz like decompositions. The minimality of the Krickeberg decomposition yields a natural ordered lattice structure on the space of convergent martingales making this space into a Dedekind complete Riesz space. Finally we show that the Riesz space of convergent martingales is Riesz isomorphic to the order closure of the union of the ranges of the conditional expectations in the filtration. Consequently we can characterize the space of order convergent martingales both in Riesz spaces and in the setting of probability spaces.     

A CHARACTERIZATION OF THE BAND OF KERNEL OPERATORS

ABSTRACT

In this note we present a characterization of the band of kernel operators in the abstract setting of Riesz spaces. Under the assumptions that E is an Archimedean Riesz space and F a Dedekind complete Riesz space separated by its ex= tended order continuo88 dual, we obtain a characterization of the band (E_oo ? F)^dd in terms of (sequentially) star or= der continuous operators.     

On the decompositions of T-quasi-martingales on Riesz spaces

The concept of a quasi-martingale is generalised to the Riesz space setting. Here we show that a quasi-martingale can be decomposed into the sum of a martingale and a quasi-potential. If, in addition, the quasi-martingale and its filtration are right continuous we show that the quasi-martingale can decomposed into the sum of a right continuous martingale and the difference of two positive right continuous potentials. The approach is measure-free and relies entirely on the order structure of Riesz spaces.     

AN f-ALGEBRA APPROACH TO THE RIESZ TENSOR PRODUCT OF ARCHIMEDEAN RIESZ SPACES

Abstract

We construct the Riesz tensor product of Archimedean Riesz spaces and derive its properties using functional calculus and f-algebras. We improve results on the approximation of elements in the Riesz tensor product by means of elements in the vector space tensor product in such a way that the order density property is a consequence of the improved approximation result.     

ON THE FUNCTIONAL CALCULUS IN ARCHIMEDEAN RIESZ SPACES WITH APPLICATIONS TO APPROXIMATION THEOREMS

ABSTRACT

We show that the functional calculus defined on the class of Dedekind σ-complete Riesz spaces can be extended to the class of uniformly complete Archimedean Riesz spaces without representing in the process the spaces involved by spaces of functions. As a consequence some results in the theory of Riesz spaces which were proved previously by representation techniques, can now be proved in an intrinsic way.     

非交换Hp空间理论的若干进展——纪念李国平院士吴新谋教授诞辰100周年

该文分两部分综述非交换H_p 空间理论的研究背景、发展线路以及某些最新进展.第一部分介绍非交换Hardy 空间理论, 包括有限次对角代数的基本性质 (如唯一正规态开拓性质、分解性质、对数模性、不变子空间性质等),Szeg\"{o} 与Riesz 型分解定理和H¹-BMO 对偶定理等. 第二部分综述非交换 H_p鞅空间理论, 主要介绍各种非交换鞅不等式以及作者与合作者在非交换Hardy 鞅空间原子分解方面获得的最新结果. 该文还给出了非交换H_p空间理论有待解决的一些问题 和潜在的发展方向.     

On Cantor-Bernstein type theorems in Riesz spaces

We generalize the main result of [21] to Riesz spaces. Let X and Y be Riesz spaces with σ-complete Boolean algebras of projection bands. If X and Y are each Riesz isomorphic to a projection band of the other space then the spaces are Riesz isomorphic. As an application of the above theorem we give an example of non-Riesz isomorphic Banach lattices such that: (1) their order (= topological) duals are Riesz isomorphic and (2) each of them is Riesz isomorphic to a projection band of the other one.     

Completion of merotopic spaces and extension of uniformly continuous maps

A general Riesz merotopic space (X, ν) determines a not necessarily topological closure operator c_ν on X. The space (X, ν) is said to be complete if every cluster on (X, ν) is contained in an adherence grill on (X, c_ν). We discuss a method of obtaining a large class of completions of a given Riesz merotopic space with induced T₁ closure space. As special cases we get the simple completion, which induces a simple closure space extension, and the strict completion, which induces a strict closure space extension. We show that the category of complete separated T₁ Riesz merotopic spaces is epireflective in the category of separated T₁ Riesz merotopic spaces, the reflection of an object being the simple completion. Similarly the category of complete clan-covered quasi-regular T₁ Riesz merotopic spaces is epireflective in the category of clan-covered quasi-regular T₁ Riesz merotopic spaces, the reflection of an object being the strict completion.     

A refinement of the Riesz decomposition for amarts and semiamarts

A real-valued adapted sequence of random variables is an amart if and only if it can be written as a sum of a martingale and a sequence dominated in absolute value by a Doob potential, i.e., a positive supermartingale that converges to 0 in L¹. The same holds for vector-valued uniform amarts with the norm replacing the absolute value.     

Adjoining an Order Unit to a Matrix Ordered Space

We prove that an order unit can be adjoined to every L ^∞-matricially Riesz normed space. We introduce a notion of strong subspaces. The matrix order unit space obtained by adjoining an order unit to an L ^∞-matrically Riesz normed space is unique in the sense that the former is a strong L ^∞-matricially Riesz normed ideal of the later with codimension one. As an application of this result we extend Arveson’s extension theorem to L ^∞-matircially Riesz normed spaces. As another application of the above adjoining we generalize Wittstock’s decomposition of completely bounded maps into completely positive maps on C ^*-algebras to L ^∞-matricially Riesz normed spaces. We obtain sharper results in the case of approximate matrix order unit spaces. Mathematics Subject Classification (2000). Primary 46L07     

鞅空间理论的新进展

本文是一篇综述性的文字,内容主要是近年来有关鞅空间理论的发展状况,特别是集中于B-值鞅和弱型鞅空间两个部分,可以看做是整个鞅空间理论发展的一个侧面.希望以此引起读者对鞅论的了解和进一步研究的兴趣.具体内容分为以下三节:1.研究鞅空间理论的意义,阐述鞅空间理论与调和分析的关系;2.鞅空间理论的向量值化,阐述B-值鞅不等式与Banach空间几何学的相互依存关系;3.弱型鞅空间与变指数鞅空间的不等式及其应用.     

Seminorms on ordered vector spaces that extend to Riesz seminorms on larger Riesz spaces

As a generalization of the notion of Riesz seminorm, a class of seminorms on directed partially ordered vector spaces is introduced, such that (1) every seminorm in the class can be extended to a Riesz seminorm on every larger Riesz space that is majorized and (2) a seminorm on an order dense linear subspace of a Riesz space is in the class if and only if it can be extended to a Riesz seminorm on the Riesz space. The latter property yields that if a directed partially ordered vector space has an appropriate Riesz completion, then a seminorm on the space is in the class if and only if it is the restriction of a Riesz seminorm on the Riesz completion. An explicit formula for the extension is given. The class of seminorms is described by means of a notion of solid unit ball in partially ordered vector spaces. Some more properties concerning restriction and extension are studied, including extension to L- and M-seminorms.     

Subspaces of Normed Riesz Spaces

It will be shown that a normed partially ordered vector space is linearly, norm, and order isomorphic to a subspace of a normed Riesz space if and only if its positive cone is closed and its norm p satisfies p(x)p(y) for all x and y with -yxy. A similar characterization of the subspaces of M-normed Riesz spaces is given. With aid of the first characterization, Krein's lemma on directedness of norm dual spaces can be directly derived from the result for normed Riesz spaces. Further properties of the norms ensuing from the characterization theorem are investigated. Also a generalization of the notion of Riesz norm is studied as an analogue of the r-norm from the theory of spaces of operators. Both classes of norms are used to extend results on spaces of operators between normed Riesz spaces to a setting with partially ordered vector spaces. Finally, a partial characterization of the subspaces of Riesz spaces with Riesz seminorms is given.     

Shift-invariant spaces on LCA groups

In this article we extend the theory of shift-invariant spaces to the context of LCA groups. We introduce the notion of H-invariant space for a countable discrete subgroup H of an LCA group G, and show that the concept of range function and the techniques of fiberization are valid in this context. As a consequence of this generalization we prove characterizations of frames and Riesz bases of these spaces extending previous results, that were known for Rd and the lattice Zd.     

An Andô-Douglas type theorem in Riesz spaces with a conditional expectation

In this paper we formulate and prove analogues of the Hahn-Jordan decomposition and an Andô-Douglas-Radon-Nikodým theorem in Dedekind complete Riesz spaces with a weak order unit, in the presence of a Riesz space conditional expectation operator. As a consequence we can characterize those subspaces of the Riesz space which are ranges of conditional expectation operators commuting with the given conditional expectation operators and which have a larger range space. This provides the first step towards a formulation of Markov processes on Riesz spaces.     

Generalised weighted composition operators on Maeda–Ogasawara spaces

Every Archimedean Riesz space can be embedded as an order dense subspace of some $C^{\infty }\left(X\right)$, the Riesz space of all extended continuous functions on a Stonean space $X$, called its Maeda–Ogasawara space. Furthermore, it is a fact that every Riesz homomorphism between spaces of ordinary continuous functions on compact Hausdorff spaces is a weighted composition operator. We prove that a generalised statement holds for Maeda–Ogasawara spaces and refine these results in case the homomorphism preserves order limits.     

On the variety of Riesz spaces

Finitely generated linearly ordered Riesz spaces are described, leading to a proof that the variety of Riesz spaces is generated as a quasivariety by the Riesz space ? of real numbers. The finitely generated Riesz spaces are also described: they are the subalgebras of real-valued function spaces on root systems of finite height.     

Some Characterizations of Riesz- Valued Sequence Spaces Generated by an Order ϕ-Function

In this paper, we introduce an order ?-function on a Riesz space. Further, we construct a Riesz- Valued Sequence spaces using the ?-function and obtain the condition to characterize these spaces.     

On differential inclusions with additive generalized functions

We study a space of potentials on the n-dimensional Euclidean space that are constructed on the basis of rearrangement-invariant spaces (RISs) by means of convolutions with kernels of general form. These spaces include the classical spaces of Bessel and Riesz potentials as particular cases. We examine the integral properties of the potentials and find necessary and sufficient conditions for their embedding in an RIS. Optimal RISs for such embeddings are also described.