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.     

On the Riesz fusion bases in Hilbert spaces

In this paper we investigate the connection between fusion frames and obtain a relation between indexes of the synthesis operators of a Besselian fusion frame and associated frame to it. Next we introduce a new notion of a Riesz fusion bases in a Hilbert space. We show that any Riesz fusion basis is equivalent with a orthonormal fusion basis. We also obtain generalizations of Theorem 4.6 of [1]. Our results generalize results obtained for Riesz bases in Hilbert spaces. Finally we obtain some results about stability of fusion frame sequences under small perturbations.     

Polynomials on Riesz spaces

In this paper we investigate polynomial mappings on Riesz spaces. We give a characterization of positivity of homogeneous polynomials in terms of forward differences. Finally we prove Hahn-Banach type extension theorems for positive and regular polynomial mappings.     

Amarts on Riesz Spaces

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.     

On decompositions of continuity in topological spaces

By using m-structures m ₁, m ₂ on a topological space (X, τ), we define a set D(m ₁,m ₂) = {A: m ₁ Int (A) = m ₂ Int (A)} and obtain many decompositions of open sets and weak forms of open sets. Then, the decompositions provide many decompositions of continuity and weak forms of continuity.     

2-Universally complete Riesz spaces and inverse-closed Riesz spaces

In this paper we show mainly two results about uniformly closed Riesz subspaces of ?^X containing the constant functions. First, for such a Riesz subspace E, we solve the problem of determining the properties that a real continuous functiondefined on a proper open interval of ?should have in order that the conditions “E is closed under composition with ” and “E is closed under inversion in X” become equivalent. The second result, reformulated in the more general frame of the Archimedean Riesz spaces with weak order unit e, establishes that E (e-uniformly complete and e-semisimple) is closed under inversion in C(Spec E) if and only if E is 2-universally e-complete.     

On homogeneous decomposition spaces and associated decompositions of distribution spaces

A new construction of decomposition smoothness spaces of homogeneous type is considered. The smoothness spaces are based on structured and flexible decompositions of the frequency space ${\mathbb{R}}^d?\{0\}$. We construct simple adapted tight frames for $L_2\left({\mathbb{R}}^d\right)$ that can be used to fully characterise the smoothness norm in terms of a sparseness condition imposed on the frame coefficients. Moreover, it is proved that the frames provide a universal decomposition of tempered distributions with convergence in the tempered distributions modulo polynomials. As an application of the general theory, the notion of homogeneous α‐modulation spaces is introduced.     

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.     

On discrete and continuous quotient Riesz spaces

Order properties of quotient Riesz spaces E/N(f) by null ideals N(f) are investigated. We show relationships between properties of a Riesz space E and its order dual E ^~ and properties of quotients E/N(f) where f runs over some subspaces of E ^~. A characterization of metrizable locally convex topological Riesz spaces whose all quotients (by proper closed ideals) are discrete is also given.     

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.     

Conditional expectations on Riesz spaces

Conditional expectations operators acting on Riesz spaces are shown to commute with a class of principal band projections. Using the above commutation property, conditional expectation operators on Riesz spaces are shown to be averaging operators. Here the theory of f-algebras is used when defining multiplication on the Riesz spaces. This leads to the extension of these conditional expectation operators to their so-called natural domains, i.e., maximal domains for which the operators are both averaging operators and conditional expectations. The natural domain is in many aspects analogous to L¹.     

Positive polynomials on Riesz spaces

We prove some properties of positive polynomial mappings between Riesz spaces, using finite difference calculus. We establish the polynomial analogue of the classical result that positive, additive mappings are linear. And we prove a polynomial version of the Kantorovich extension theorem.     

Markov processes on Riesz spaces

Measure-free discrete time stochastic processes in Riesz spaces were formulated and studied by Kuo, Labuschagne and Watson. Aspects relating martingales, stopping times, convergence of these processes as well as various decomposition were considered. Here we formulate and study Markov processes in a measure-free Riesz space setting.     

On decompositions of Banach spaces of continuous functions on Mrówka's spaces

It is well known that if is infinite compact Hausdorff and scattered (i.e., with no perfect subsets), then the Banach space of continuous functions on has complemented copies of , i.e., . We address the question if this could be the only type of decompositions of into infinite-dimensional summands for infinite, scattered. Making a special set-theoretic assumption such as the continuum hypothesis or Martin's axiom we construct an example of Mrówka's space (i.e., obtained from an almost disjoint family of sets of positive integers) which answers positively the above question.

     

Riesz decompositions and subtractivity for excessive measures

The convex cone of excessive measures of a right Markov process is an example of a superharmonic semigroup in the abstract potential theory developed by Arsove and Leutwiler. We show that their theory of Riesz decompositions can be sharpened in the case of excessive measures. In particular there is always a Riesz decomposition relative to a given potential cone (resp. harmonic cone). An element of an ordered convex cone is subtractive if each majorant is a specific majorant. This notion of subtractivity features prominently in the theory of harmonic cones. We give a complete characterization of the subtractive elements in the cone of excessive measures.The research of both authors was supported in part by NSF Grant DMS 87-21347.     

On the distribution function with respect to conditional expectation on Riesz spaces

The notion of distribution function with respect to a conditional expectation is defined and studied in the framework of Riesz spaces.     

On Markuševič decompositions of Banach spaces

пРОВЕДЕНО сИстЕМАтИ ЧЕскОЕ ИсслЕДОВАНИЕ РАжлИЧНых тИпОВ РАжлОжЕНИИ, А ИМ ЕННО, пО МАРкУшЕВИЧУ, ОБОБЩЕН Ных И РАжлОжЕНИИ пО шА УДЕРУ. УкАжАНы пРИМЕРы, ИллУ стРИРУУЩИЕ ВжАИМООтНОшЕНИь ЁтИ х тИпОВ РАжлОжЕНИИ. Ос НОВНАь цЕль ДАННОИ РАБОты — ИжУЧЕ НИЕ РАжлОжЕНИИ пО МАРкУшЕВИЧУ (M-РАжл ОжЕНИИ). ОтМЕЧАЕтсь, Чт О пОслЕДОВАтЕльНОсть пРОЕкцИИ, сВьжАННАь с ОБОБЩЕНН ыМ РАжлОжЕНИЕМ, ьВльЕ тсь ЕДИНстВЕННОИ тОгДА И тОлькО тОгДА, кОгДА ЁтОM-РАжлОжЕНИЕ. пОлУ ЧЕНА хАРАктЕРИжАцИьM-РАжлОжЕНИИ В тЕРМИН Ах Их пОДпОслЕДОВАтЕ льНОстЕИ. пОлУЧЕНА тАкжЕ хАРАк тЕРИжАцИь пО шАУДЕРУ В тЕРМИНАхM-РАжлОжЕНИИ. НАкОНЕц, пОлУЧЕНы ОтНОшЕНИь Д ВОИстВЕННОстИ МЕжДУM-РАжлОжЕНИьМИ НЕкОтО РОгО пРОстРАНстВА И НЕкОтОРыМИ РАжлОжЕ НИьМИ сОпРьжЕННОгО п РОстРАНстВА.     

Probabilistic normed Riesz spaces

In this paper, the concepts of probabilistic normed Riesz space and probabilistic Banach lattice are introduced, and their basic properties are studied. In this context, some continuity and convergence theorems are proved.     

Discrete-time stochastic processes on Riesz spaces

It has been recognised that order is closely linked with probability theory, with lattice theoretic approaches being used to study Markov processes but, to our knowledge, the complete theory of (sub, super) martingales and their stopping times has not been formulated on Riesz spaces. We generalize the concepts of stochastic processes, (sub, super) martingales and stopping times to Riesz spaces. In this paper we consider discrete time processes with bounded stopping times.