Vector Lattices Research Articles

Order convergence is not topological in infinite-dimensional vector lattices

Order convergence is not topological in infinite-dimensional vector lattices

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.

Valuations: From orthogonal additivity to orthosymmetry

We prove that polynomial valuations on vector lattices correspond to orthosymmetric multilinear maps. As a consequence we obtain a concise proof of the correspondence between orthosymmetry and orthogonal additivity.

Vector Lattices Admitting a Positively Homogeneous Continuous Function Calculus

Abstract We characterize the Archimedean vector lattices that admit a positively homogeneous continuous function calculus by showing that the following two conditions are equivalent for each $n$-tuple $\boldsymbol{x} = (x_1,\ldots ,x_n)\in X^n$, where $X$ is an Archimedean vector lattice and $n\in{\mathbb{N}}$: • there is a vector lattice homomorphism $\Phi _{\boldsymbol{x}}\colon H_n\to X$ such that $$\begin{equation*}\Phi_{\boldsymbol{x}}(\pi_i^{(n)}) = x_i\qquad (i\in\{1,\ldots,n\}),\end{equation*}$$where $H_n$ denotes the vector lattice of positively homogeneous, continuous, real-valued functions defined on ${\mathbb{R}}^n$ and $\pi _i^{(n)}\colon{\mathbb{R}}^n\to{\mathbb{R}}$ is the $i^{\text{}}$th coordinate projection;• there is a positive element $e\in X$ such that $e\geqslant \lvert x_1\rvert \vee \cdots \vee \lvert x_n\rvert$ and the norm$$\begin{equation*}\lVert x\rVert_e = \inf\bigl\{ \lambda\in[0,\infty)\:\colon\:\lvert x\rvert{\leqslant}\lambda e\bigr\},\end{equation*}$$defined for each $x$ in the order ideal $I_e$ of $X$ generated by $e$, is complete when restricted to the closed sublattice of $I_e$ generated by $x_1,\ldots ,x_n$. Moreover, we show that a vector space which admits a ‘sufficiently strong’ $H_n$-function calculus for each $n\in{\mathbb{N}}$ is automatically a vector lattice, and we explore the situation in the non-Archimedean case by showing that some non-Archimedean vector lattices admit a positively homogeneous continuous function calculus, while others do not.

Solvability of the system of implicit generalized order complementarity problems

In this paper, we introduce the notion of exceptional family for the system of implicit generalized order complementarity problems in vector lattice. We present some alternative existence results of the solutions for the system of implicit generalized order complementarity problems via topological degree aspects. The new developments in this paper generalize and improve some known results in the literature.

Асимптотическое описание быстрых тепловых процессов в скалярных гармонических решетках

Thermal processes in d-dimensional (d = 1,2) scalar harmonic lattices with simple structure are considered. The redistribution between the averaged kinetic and potential energies of particles after instantaneous thermal excitation is described. At times much longer than the characteristic period of atomic oscillations, the difference between the kinetic and potential energies oscillates with amplitude decreasing by a power law (typically, the exponent is -d/2). The frequencies of the oscillations are determined from the dispersion relation for the lattice. The above results can be generalized for scalar and vector lattices of complex structure and various dimensions. As an example, the oscillations of the kinetic tmperature in the two-dimensional hexagonal lattice (graphene lattice) are investigated.

Some Notes on b-Weakly Compact Operators

The main aim of this paper is studying the family $$W_b(E,F)$$ of b-weakly compact operators between two Banach lattices. For an order dense sublattice G of a vector lattice E, if $$T:G\rightarrow F$$ is a b-weakly compact operator between two Banach lattices, then $$T\in W_b(E,F)$$ whenever the norm of E is order continuous and $$T:E\rightarrow F$$ is a positive operator. We also investigate the relationship between $$W_b(E,F)$$ and some other classes of operators like $$L^{(1)}_c(E,F)$$ and $$L^{(2)}_c(E,F)$$ .

Unbounded Order Convergence and the Gordon Theorem

The celebrated Gordons theorem is a natural tool for dealing with universal completions of Archimedean vector lattices. Gordons theorem allows us to clarify some recent results on unbounded order convergence. Applying the Gordon theorem, we demonstrate several facts on order convergence of sequences in Archimedean vector lattices. We present an elementary Boolean-Valued proof of the Gao--Grobler--Troitsky--Xanthos theorem saying that a sequence x_n in an Archimedean vector lattice X is uo-null (uo-Cauchy) in X if and only if x_n is o-null (o-convergent) in Xu. We also give elementary proof of the theorem, which is a result of contributions of several authors, saying that an Archimedean vector lattice is sequentially uo-complete if and only if it is sigma-universally complete. Furthermore, we provide a comprehensive solution to Azouzis problem on characterization of an Archimedean vector lattice in which every uo-Cauchy net is o-convergent in its universal completion.

He Gordon Theorem: Origins and Meaning

Boolean valued analysis, the term coined by Takeuti, signifies a branch of functional analysis which uses a special technique of Boolean valued models of set theory. The fundamental result of Boolean valued analysis is Gordons Theorem stating that each internal field of reals of a Boolean valued model descends into a universally complete vector lattice. Thus, a remarkable opportunity opens up to expand and enrich the mathematical knowledge by translating information about the reals to the language of other branches of functional analysis. This is a brief overview of the mathematical events around the Gordon Theorem. The relationship between the Kantorovichs heuristic principle and Boolean valued transfer principle is also discussed.

Complementation in the Fremlin vector lattice symmetric tensor products-II

For a vector lattice E and $$n \in \mathbb {N}$$, let $${\bar{\otimes }}_{n,s}E$$ denote the n-fold Fremlin vector lattice symmetric tensor product of E. For $$m, n \in \mathbb {N}$$ with $$m > n$$, we prove that (i) if $${\bar{\otimes }}_{m,s}E$$ is uniformly complete then $${\bar{\otimes }}_{n,s}E$$ is positively isomorphic to a complemented subspace of $${\bar{\otimes }}_{m,s}E$$, and (ii) if there exists such that $$\ker (\phi )$$ is a projection band in E then $${\bar{\otimes }}_{n,s}E$$ is lattice isomorphic to a projection band of $${\bar{\otimes }}_{m,s}E$$. We also obtain analogous results for the n-fold Fremlin projective symmetric tensor product $${\hat{\otimes }}_{n,s,|\pi |}E$$ of E where E is a Banach lattice.

Atomic sublattices and basic derivatives in finance

Suppose that E is a vector lattice where the ordering and the lattice operations in E are defined pointwise by a countable family $${\mathcal {F}}=\{f_i|i\in {{\mathbf {N}}}\}$$ of positive linear functional of E and Z is a sublattice of E. Based on algebraic and order properties of E we give necessary and sufficient conditions in order Z to be atomic. Especially we show the existence of a basic sequence $$\{b_n\}$$ of extremal points (atoms) of $$Z_+$$ so that for any $$x\in Z_+$$ a unique sequence $$({\widehat{x}}(n))$$ of real components of x with respect to $$\{b_n\}$$ exists so that $$x=\sup \{{\widehat{x}}(n)b_n\;|\;n\in {{\mathbf {N}}}\}$$ and also $$x=sup_{n}\sum _{i=1}^n{\widehat{x}}(i)b_i$$ . These results give an answer to the problem of the existence of basic derivatives in financial markets.

Order Continuous Operators on Pre-Riesz Spaces and Embeddings

We investigate properties of order continuous operators on pre-Riesz spaces with respect to the embedding of the range space into a vector lattice cover or, in particular, into its Dedekind completion. We show that order continuity is preserved under this embedding for positive operators, but not in general. For the vector lattice \ell_0^\infty of eventually constant sequences, we consider the pre-Riesz space of regular operators on \ell_0^\infty and show that making the range space Dedekind complete does not provide a vector lattice cover of the pre-Riesz space. A similar counterexample is obtained for the directed part of the space of order continuous operators on \ell_0^\infty .

Two Applications of Boolean Valued Analysis

The paper contains two main results that are obtained by Boolean valued analysis. The first asserts that a universally complete vector lattice without locally one-dimensional bands can be decomposed into a direct sum of two vector sublattices that are laterally complete and invariant under all band projections and there exists a band preserving linear isomorphism of each of these sublattices to the original lattice. The second result establishes a counterpart of the Ando Theorem on the joint characterization of $A\!L^p$ and $c_0$ for the class of cyclic Banach lattices, using the Boolean valued transfer for injective Banach lattices.

Narrow and C-compact Orthogonally Additive Operators in Lattice-Normed Spaces

In this article we consider orthogonally additive operators on lattice-normed spaces. In the first part of the article we present some examples of narrow, laterally-to-norm continuous and C-compact operators defined on a lattice-normed space and taking value in a Banach space. We show that any laterally-to-norm continuous narrow orthogonally additive operator defined on a decomposable lattice-normed space (V, E) over an atomic vector lattice E with the projection property is equal to zero. In the second part we prove that the sum of two orthogonally additive operators $$T+S$$ defined on a order complete, decomposable lattice-normed space V and taking value in Banach space X, where $$T:V\rightarrow X$$ is a laterally-to-norm continuous C-compact operator and $$S:V\rightarrow X$$ is a narrow operator, is a narrow operator as well.

Optical vector lattice breathers of a two-component Rabi-coupled Gross–Pitaevskii system with variable coefficients

A two-component Rabi-coupled Gross–Pitaevskii system with variable coefficients is under investigation. The system can be used to describe the optical vector pulse propagations in two complex electric fields in Bose–Einstein condensates. A rotation transformation and a similarity transformation are successively implemented to transform the system into the celebrated Manakov system. Via utilizing the vector breather solutions obtained of the Manakov system, and choosing several typical kinds of similarity transformation functions, novel vector lattice breathers are excited and observed. These breathers possess lattice background waves, and exhibit abundant patterns.

An almost orthosymmetric bilinear map

In this paper, as a generalization of the concept of pseudo-almost f -algebra, we define a new concept of almost orthosymmetric bilinear map on a vector lattice and prove that the Arens triadjoint of a positive almost orthosymmetric bilinear map is positive almost orthosymmetric.This also extends results on the order bidual of pseudo-almost f -algebras.

Unbounded p-Convergence in Lattice-Normed Vector Lattices

A net xα in a lattice-normed vector lattice (X, p, E) is unbounded p-convergent to x ∈ X if $$p\left({\left| {{x_\alpha} - x} \right| \wedge u} \right)\buildrel o \over \longrightarrow $$ 0 for every u ∈ X+. This convergence has been investigated recently for (X, p, E) = (X, |·|, X) under the name of uo-convergence, for (X, p, E) = (X, ‖·‖, ℝ) under the name of un-convergence, and also for $$\left({X,\;p,\;{\mathbb{R}^{X^{\prime}}}} \right)$$ , where p(x)[f]:= |f|(|x|), under the name uaw-convergence. In this paper we study general properties of the unbounded p-convergence.

On order continuous duals of vector lattices of continuous functions

A topological space X is called resolvable if it contains a dense subset with dense complement. Using only basic principles, we show that whenever the space X has a resolving subset that can be written as an at most countably infinite union of subsets, in such a way that a given vector lattice of (not necessarily bounded) continuous functions on X separates every point outside the resolving subset from each of its constituents, then the order continuous dual of this lattice is trivial. In order to apply this result in specific cases, we show that several spaces have resolving subsets that can be written as at most countably infinite unions of closed nowhere dense subsets. An appeal to the main result then yields, for example, that, under appropriate conditions, vector lattices of continuous functions on separable spaces, metric spaces, and topological vector spaces have trivial order continuous duals if they separate points and closed nowhere dense subsets. Our results in this direction extend known results in the literature. We also show that, under reasonably mild separation conditions, vector lattices of continuous functions on locally connected T1 Baire spaces without isolated points have trivial order continuous duals. A discussion of the relation between our results and the non-existence of non-zero normal measures is included.

Complementation in the Fremlin vector lattice symmetric tensor products-I

For a positive integer n and a uniformly complete vector lattice E, let denote the n-fold Fremlin vector lattice symmetric tensor product of E. We prove that if there exists a lattice homomorphism φ ∈ E ∼ then E is lattice isomorphic to a complemented sublattice of . Moreover, if ker(φ) is a projection band in E then the image of E is also a projection band in .

Limit laws for martingales in vector lattices

We prove limit laws for martingales in vector lattices, under suitable technical hypotheses involving the increments of the sequences therein.