Topological Vector Lattices Research Articles

Locally solid convergences and order continuity of positive operators

We consider vector lattices endowed with locally solid convergence structures, which are not necessarily topological. We show that such a convergence is defined by the convergence to 0 on the positive cone. Some results on unbounded modification which were only available in partial cases are generalized. Order convergence is characterized as the strongest locally solid convergence in which monotone nets converge to their extremums (if they exist). We partially characterize sublattices on which the order convergence is the restriction of the order convergence on the ambient lattice. We prove that a homomorphism is order continuous iff it is uo-continuous. Uo convergence is characterized independently of order convergence. We show that on the space of continuous function uo convergence is weaker than the compact open convergence iff the underlying topological space contains a dense locally compact subspace. For a large class of convergences we prove that a positive operator is order continuous if and only if its restriction to a dense regular sublattice is order continuous, and that the closure of a regular sublattice is regular with the original sublattice being order dense in the closure. We also present an example of a regular sublattice of a locally solid topological vector lattice whose closure is not regular.

Locally convex topological vector lattices and their representation

Locally convex topological vector lattices and their representation

$$o\tau $$-Continuous, Lebesgue, KB, and Levi Operators Between Vector Lattices and Topological Vector Spaces

We investigate $$o\tau $$ -continuous/bounded/compact and Lebesgue operators from vector lattices to topological vector spaces; the Kantorovich-Banach operators between locally solid lattices and topological vector spaces; and the Levi operators from locally solid lattices to vector lattices. The main idea of operator versions of notions related to vector lattices lies in redistributing topological and order properties of a topological vector lattice between the domain and range of an operator under investigation. Domination properties for these classes of operators are studied.

A geometric Vietoris-Begle theorem, with an application to convex subsets of topological vector lattices

We show that if L is a topological vector lattice, u:L→L is the function u(x)=x∨0, C⊂L is convex, and D=u(C) is metrizable, then D is an ANR and u|C:C→D is a homotopy equivalence, so D is contractible and thus an AR. This is proved by verifying the hypotheses of a second result: if X is a connected space that is homotopy equivalent to an ANR, Y is an ANR, and f:X→Y is a continuous surjection such that, for each y∈Y and each neighborhood V⊂Y of y, there is a neighborhood V′⊂V of y such that f−1(V′) can be contracted in f−1(V), then f is a homotopy equivalence. The latter result is a geometric analogue of the Vietoris-Begle theorem.

No-arbitrage concepts in topological vector lattices

We provide a general framework for no-arbitrage concepts in topological vector lattices, which covers many of the well-known no-arbitrage concepts as particular cases. The main structural condition we impose is that the outcomes of trading strategies with initial wealth zero and those with positive initial wealth have the structure of a convex cone. As one consequence of our approach, the concepts NUPBR, NAA_1 and NA_1 may fail to be equivalent in our general setting. Furthermore, we derive abstract versions of the fundamental theorem of asset pricing (FTAP), including an abstract FTAP on Banach function spaces, and investigate when the FTAP is warranted in its classical form with a separating measure. We also consider a financial market with semimartingales which does not need to have a numéraire, and derive results which show the links between the no-arbitrage concepts by only using the theory of topological vector lattices and well-known results from stochastic analysis in a sequence of short proofs.

Extension of Positive Operators

The main result states that if E is a separable Frechet lattice and F is a (locally solid) topological vector lattice with the σ-interpolation property then each positive linear operator T0 from a majorizing subspace G ⊂ E into F admits extension to a continuous positive linear operator T from E into F. This fact is proved by using only the axiom of countable choice.

Monotone countable paracompactness and maps to ordered topological vector spaces

In this paper, we characterize monotonically countably paracompact (or monotonically countably metacompact) spaces by semi-continuous maps with values into some ordered topological vector spaces or some topological vector lattices. These extend earlier results for real-valued semi-continuous functions.

Abstract convex approximations of nonsmooth functions

In the article we use abstract convexity theory in order to unify and generalize many different concepts of nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract convex (concave) approximations of a nonsmooth function mapping a topological vector space to an order complete topological vector lattice. We study basic properties of these notions, construct elaborate calculus of abstract codifferentiable functions and discuss continuity of abstract codifferential. We demonstrate that many classical concepts of nonsmooth analysis, such as subdifferentiability and quasidifferentiability, are particular cases of the concepts of abstract codifferentiability and abstract quasidifferentiability. We also show that abstract convex and abstract concave approximations are a very convenient tool for the study of nonsmooth extremum problems. We use these approximations in order to obtain various necessary optimality conditions for nonsmooth nonconvex optimization problems with the abstract codifferentiable or abstract quasidifferentiable objective function and constraints. Then, we demonstrate how these conditions can be transformed into simpler and more constructive conditions in some particular cases.

Recent Development in Fixed-Point Theory, Optimization, and Their Applications

This special volume was originally conceived to provide authors and readers a publication to present the most recent advances in the study of applications of various fixed point theorems. Since 1909, when Luitzen Brouwer proved the first fixed point theorem named after him, fixed point theory has played very important roles in many different fields. We can find a lot of demonstrations in optimization theory, approximation theory, differential equations, variational inequalities, complementary problems, equilibrium theory, game theory, economics theory, and so forth. Fixed point theorems are developed for single-valued or set-valued mappings of metric spaces, topological vector spaces, posets and lattices, Banach lattices, . . .. Among the themes of fixed point theory, the topic of approximation of fixed points of mappings is particularly important because it is useful for proving the existence of fixed points ofmappings. It can be applied to prove the solvability of optimization problems, differential equations, variational inequalities, and equilibrium problems. Due to the importance of and the high volume of active research in the nonlinear analysis and optimization and, in particular, many new tools in studying them involving fixed point approximations, it is worthwhile to publish a special issue on this topic to highlight the recent advances in this field. The selection of the papers included in this volume, which has been based on a strict international peer review procedure, contains a representative list of papers with newfangled results which covers the different topics considered in our original proposal. In the papers in this special issue, the following topics are discussed.

Bartle–Dunford–Schwartz integration

We study the BDS-integral, using the original definition, but with respect to a convexly bounded measure μ with values in an arbitrary sequentially complete tvs X. Denote by L0(μ) the space of μ-measurable R-valued functions. Then all bounded measurable functions are μ-integrable (in an elementary sense), and the space L1(μ) of BDS-integrable functions is a vector lattice and a topological vector space in its natural topology. We next distinguish the space L∘1(μ) as the largest vector subspace of L1(μ) that is solid in L0(μ). We prove general convergence theorems for both L1(μ) and L∘1(μ). In particular, we show that L∘1(μ) with its natural topology is a Dedekind σ-complete topological vector lattice with the σ-Lebesgue property, and that the Dominated Convergence Theorem holds in L∘1(μ). If X contains no isomorphic copy of c0, then L∘1(μ) has the σ-Levi property (that is, the Beppo Levi Theorem holds). We identify L∘1(μ) as the domain for the Thomas–Turpin integral, and thus show that this integral is simply the restriction of the BDS-integral to L∘1(μ). The last two statements answer the questions posed, respectively, by Thomas and Turpin, and left open since the seventies.

On nestedly complete topological vector lattices

A topological vector lattice E is called (σ-)nestedly complete if every downward directed net (resp., decreasing sequence) of order intervals in E whose ‘diameters’ tend to zero has a nonempty intersection. Some characterizations of the (σ-)nested completeness are given, and it is shown that if E is metrizable and nestedly complete, so is each of its quotients E/I, where I is a closed ideal in E. Conversely, if a closed ideal I in E is (sequentially) complete and E/I is (σ-)nestedly complete, so is E. However, the nested completeness is not a three-space property: an example is given where both I and E/I are nestedly complete while E is not. It is also shown that the nested completeness and the related notion of nested density come up quite naturally when extending some positive linear operators. Finally, the nested and other completeness type properties of vector lattices C(S) are investigated.

Insertion theorems for maps to Banach lattices

We investigate semi-continuous maps from topological spaces into topological vector lattices. As an application, several insertion theorems with values in Banach lattices are given.

Banach–Stone Theorems for maps preserving common zeros

Let X and Y be completely regular spaces and E and F be Hausdorff topological vector spaces. We call a linear map T from a subspace of C(X, E) into C(Y, F) a Banach–Stone map if it has the form T f (y) = S y (f (h(y))) for a family of linear operators S y : E → F, $${y \in Y}$$ , and a function h: Y → X. In this paper, we consider maps having the property: $$\bigcap^{k}_{i=1}Z(f_{i}) \neq\emptyset \iff \bigcap^{k}_{i=1}Z(Tf_{i})\neq\emptyset , \quad({\rm Z}) $$ where Z(f) = {f = 0}. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces of differentiable functions (including C ∞), as Banach–Stone maps. In particular, we confirm a conjecture of Ercan and Onal: Suppose that X and Y are realcompact spaces and E and F are Hausdorff topological vector lattices (respectively, C *-algebras). Let T: C(X, E) → C(Y, F) be a vector lattice isomorphism (respectively, *-algebra isomorphism) such that $$ Z(f) \neq\emptyset\iff Z(Tf) \neq\emptyset. $$ Then X is homeomorphic to Y and E is lattice isomorphic (respectively, C *-isomorphic) to F. Some results concerning the continuity of T are also obtained.

Equilibrium theory with asymmetric information and with infinitely many commodities

The traditional deterministic general equilibrium theory with infinitely many commodities cannot cover economies with private information constraints on the consumption sets. We bring the level of asymmetric information equilibrium theory at par with that of the deterministic one. In particular, we establish results on equilibrium existence for exchange economies with asymmetric (differential) information and with an infinite dimensional commodity space. Our new equilibrium existence theorems include, as a special case, classical results, e.g. Bewley [Existence of equilibria in economies with infinitely many commodities, J. Econ. Theory 4 (1972) 514–540] or Mas-Colell [The price equilibrium existence problem in topological vector lattices, Econometrica 54 (1986) 1039–1053].

An isomorphic characterization of L1-spaces

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 + Y + = ℝ + B ¯ ( u , 1 ) (for ‖ u‖ >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials.

S-strictly quasi-concave vector maximisation

In this paper, we discuss the relationship among the concepts of an S-strictly quasiconcave vector-valued function introduced by Benson and Sun, a C-strongly quasiconcave vector-valued function and a C-strictly quasiconcave vector-valued function in a topological vector space with a lattice ordering. We generalize a main result obtained by Benson and Sun about the closedness of an efficient solution set in multiple objective programming. We prove that an efficient solution set is closed and connected when the objective function is a continuous S-strictly quasiconcave vector-valued function, the objective space is a topological vector lattice and the ordering cone has a nonempty interior.

Localisation of continuity to bounded sets for nonmetrisable vector topologies and its applications to economic equilibrium theory

We present two ‘localisation’ techniques which facilitate verifications of the topological properties of sets, functions and correspondences needed in, e.g., economic equilibrium analysis with infinite-dimensional commodity spaces. The first uses the Krein-Smulian theorem, which shows that weak ∗ upper semicontinuity of a concave function on a dual Banach space is equivalent to bounded weak ∗ u.s. continuity. The second is based on the continuity of lattice operations: for a nondecreasing function on a topological vector lattice, we show that lower semicontinuity on a set bounded from below is equivalent to l.s. continuity on bounded subsets. In the case of L ∞, we use convergence in measure to establish that sequential semicontinuity, lower for the Mackey topology or upper for the weak ∗ one, is equivalent to semicontinuity. This greatly simplifies some arguments; e.g., the Mackey continuity of a concave, nondecreasing integral functional on L + ∞ becomes an immediate consequence of Lebesgue's Dominated Convergence Theorem. Other uses in mathematical economics are also discussed.

Production equilibria in locally proper economies with unbounded and unordered consumers

We prove a theorem on the existence of general equilibrium for a production economy with unordered preferences in a topological vector lattice commodity space. Our consumption sets need not have a lower bound and the set of feasible allocations need not be topologically bounded. Instead, we introduce a notion of local proper dominance and assume that the set of feasible allocations not locally properly dominated by any other feasible allocation has compact closure. Furthermore, we assume that the economy is locally proper as opposed to uniformly proper. In particular, preferences satisfy a locally uniform version of the extreme desirability condition of Yannelis and Zame [Yannelis, N.C., Zame, W.R., 1986, Equilibria in Banach lattices without ordered preferences, Journal of Mathematical Economics 15, 85–110.].

A New Approach to the Limit Theorem on the Core of an Economy in Vector Lattices

Aliprantis, Brown, and Burkinshaw (Econometrica55(1987), 1109–1137) proved that Edgeworth equilibria for exchange economies in topological vector lattices are pseudo-equilibria. In this paper we extend their result to more general spaces. We consider pure trade economies in a vector lattice commodity space with a lattice ordered price space. By adapting the techniques of Mas-Colell and Richard (J. Econ. Theory53(1991), 1–11) we obtain the required decentralizing continuous price as the supremum of tangent functionals that support the sets of preferred net trades.Journal of Economic LiteratureClassification Numbers: C62, C71, D51.

An extension of the Bochner integral generalizing the Loeb–Osswald integral

In [10] Loeb and Osswald develop a Daniell–Stone integration theory in topological vector lattices. As an application they obtain an extension of the Bochner integral for functions with values in a topological vector lattice. The Daniell–Stone approach requires an order structure for definition of a null function. Here we build on the ideas in [10], but abandon the lattice structure. The internal representatives of the extended integrable functions in [10] turn out to be S-integrable liftings. This is our starting point: we define the class of extended integrable functions as the class of functions that have S-integrable liftings; a definition that is independent of any order structure. As a consequence, several of the proofs in the construction of the Banach space of extended integrable functions are almost identical to the analogous proofs in [10] and we omit them. After defining the integral we show that each extended integrable function induces a countably additive vector measure of bounded variation. This permits us to use the tools of the theory of vector measures. If the nonstandard hull of the Banach space has the Radon–Nikodym property, then the vector measure induced by an extended integrable function has a Bochner integrable Radon–Nikodym derivative; this yields a characterization of the Banach space of extended integrable functions. On the other hand for some spaces we can construct internal functions and standardize them to get very ‘spiky’ extended integrable functions whose induced measure is not differentiable on any set of positive measure. With this method we show that if a Banach space is not superreflexive, then it does not have the super-Radon–Nikodym property.