Vector Lattices Research Articles

A topology on the Fremlin tensor product of locally convex-solid vector lattices

A topology on the Fremlin tensor product of locally convex-solid vector lattices

Unbounded Star Convergence in Lattices

Let L be a vector lattice, "(" x_α ") " be a L-valued net, and x∈L . If |x_α-x|∧u→┴o 0 for every u ∈〖 L〗_+ then it is said that the net "(" x_α ")" unbounded order converges to x and is denoted by □(x_α □(→┴uo x)) . This definition of unbounded order convergence has been extensively studied on many structures, including vector lattices, local solid vector lattices, normed lattices and lattice normed spaces. It is not possible to apply this type of convergence to general lattices due to the lack of algebraic structure. Therefore, we will use a type of convergence that is considered to be the motivation for this type of convergence, first defined as independent convergence in semi-ordered linear spaces and later called unbounded order convergence. Namely, L is a lattice, x_α is an L -valued net, and x ϵ L . If (x_α∧b )∨a order converges to (x∧b )∨a for every a,b∈L with a≤b, then it is said that "(" x_α ")" individual converges to x or unbounded order converges to x . This definition can be easily applied to general lattices. In this article, this definition will be understood as unbounded order convergence. Also, even if these two convergences are called by the same name, there is no equivalence between them for general lattices, an example of this is mentioned in this article. Let L be a partially ordered set, "(" x_α ")" be an L -valued net and x∈L (x_α) is said to be star convergent to x if every subnet of the net (x_α ) has a subnet that is order convergent to x and denoted by x_α □(→┴s x). In this paper, a new type of convergence on lattices is defined by combining unbounded order convergence (individual convergence) and star convergence. Let L be a lattice, (x_α ) a net and x∈L (x_α) is said to be unbounded star convergent to x if for every subnet (x_β) of (x_α), there exists a subnet (x_ζ) of (x_β) such that (x_ζ∧b)∨ □(a→┴o ) (x∧b)∨a for every a,b∈L with a≤b and it is denoted by x_α □(→┴us x). The differences between the new type of convergence, called unbounded star convergence, and order convergence, star convergence are demonstrated with counterexamples. The meaningfulness of the unbounded star convergence type is analyzed with these counterexamples and the implications presented. In addition, basic questions about unbounded star convergence of a given net on lattices such as convergence of a fixed net, uniqueness of the limit, convergence of the subnet of a convergent net are answered.

Multimorphisms on pre-Riesz spaces of continuous functions

In the theory of vector lattices, mostly three classes of lattice ordered algebras are investigated, called f-algebras, d-algebras, and almost-f-algebras. Each class is endowed with a different type of positive bilinear map as the multiplication, namely with bi-orthomorphisms, bi-Riesz homomorphisms, and orthosymmetric maps, respectively. On the space C(K), with K a compact Hausdorff space, representations as weighted composition or integral operators are known. We give generalized definitions of the above mentioned maps in the setting of partially ordered vector spaces, where we deal with n-linear maps. We generalize the representations to the case of certain subspaces of some C(K) space, with K a compact Hausdorff space. With the representations at hand, we deduce basic properties of these maps. The results also cover the case of order unit spaces, which can be seen as order dense subspaces of some C(K) space via the functional representation.

Monotone-Cevian and Finitely Separable Lattices

AbstractA distributive lattice with zero is completely normal if its prime ideals form a root system under set inclusion. Every such lattice admits a binary operation $$(x,y)\mapsto x\mathbin {\smallsetminus }y$$ ( x , y ) ↦ x \ y satisfying the rules $$x\le y\vee (x\mathbin {\smallsetminus }y)$$ x ≤ y ∨ ( x \ y ) and $$(x\mathbin {\smallsetminus }y)\wedge (y\mathbin {\smallsetminus }x)=0$$ ( x \ y ) ∧ ( y \ x ) = 0 — in short a deviation. In this paper we study the following additional properties of deviations: monotone (i.e., isotone in x and antitone in y) and Cevian (i.e., $$x\mathbin {\smallsetminus }z\le (x\mathbin {\smallsetminus }y)\vee (y\mathbin {\smallsetminus }z)$$ x \ z ≤ ( x \ y ) ∨ ( y \ z ) ). We relate those matters to finite separability as defined by Freese and Nation. We prove that every finitely separable completely normal lattice has a monotone deviation. We pay special attention to lattices of principal $$\ell $$ ℓ -ideals of Abelian $$\ell $$ ℓ -groups (which are always completely normal). We prove that for free Abelian $$\ell $$ ℓ -groups (and also free $$\Bbbk $$ k -vector lattices) those lattices admit monotone Cevian deviations. On the other hand, we construct an Archimedean $$\ell $$ ℓ -group with strong unit, of cardinality $$\aleph _1$$ ℵ 1 , whose principal $$\ell $$ ℓ -ideal lattice does not have a monotone deviation.

Orthogonally additive operators on complex vector lattices

In this article we continue investigation of the lateral order on complex vector lattices started in [10,39]. We extend some of main results of [35] and [32] to the setting of operators on complex vector lattices. We establish the Riesz-Kantorovich type calculus for regular orthogonally additive operators defined on the complexification EC of an uniformly complete vector lattice E with the principal projection property and taking values in Dedekind complete vector lattice F. We prove that a regular orthogonally additive operator T:EC→F from the complexfication EC of an uniformly complete vector lattice E with the principal projection property to a Banach lattice F with an order continuous norm is narrow if and only if the modulus |T|:EC→F is.

Simpler characterizations of total orderization invariant maps

Given a finite subset [Formula: see text] of a distributive lattice, its total orderization [Formula: see text] is a natural transformation of [Formula: see text] into a totally ordered set. Recently, the author showed that multivariate maps on distributive lattices which remain invariant under total orderizations generalize various maps on vector lattices, including bounded orthosymmetric multilinear maps and finite sums of bounded orthogonally additive polynomials. Therefore, a study of total orderization invariant maps on distributive lattices provides new perspectives for maps widely researched in vector lattice theory. However, the unwieldy notation of total orderizations can make calculations extremely long and difficult. In this paper we resolve this complication by providing considerably simpler characterizations of total orderization maps. Utilizing these easier representations, we then prove that a lattice multi-homomorphism on a distributive lattice is total orderization invariant if and only if it is symmetric, and we show that the diagonal of a symmetric lattice multi-homomorphism is a lattice homomorphism, extending known results for orthosymmetric vector lattice homomorphisms.

Differentiable, holomorphic, and analytic functions on complex Φ-algebras

Using the notion of order convergent nets, we develop an order-theoretic approach to differentiable functions on Archimedean complex Φ-algebras. Most notably, we improve the Cauchy-Hadamard formulas for universally complete complex vector lattices given by both authors in a previous paper in order to prove that analytic functions are holomorphic in this abstract setting.

Free uniformly complete vector lattices

Free uniformly complete vector lattices

On Extreme Extension of Positive Operators

Given vector lattices $E$, $F$ and a positive operator $S$ from a majorzing subspace $D$ of $E$ to~$F$, denote by $\mathcal{E}(S)$ the collection of all positive extension of $S$ to all of $E$. This note aims to describe the collection of extreme points of the convex set $\mathcal{E}(T\circ S)$. It is proved, in particular, that $\mathcal{E}(T\circ S)$ and $T\circ\mathcal{E}(S)$ coincide and every extreme point of $\mathcal{E}(T\circ S)$ is an~extreme point of $T\circ\mathcal{E}(S)$, whenever $T:F\to G$ is a~Maharam operator between Dedekind complete vector lattices. The proofs of the main results are based on three ingredients: a characterization of extreme points of subdifferentials, abstract disintegration in~Kantorovich spaces, and an intrinsic characterization of subdifferentials.

A structure theorem for truncations on an Archimedean vector lattice

A structure theorem for truncations on an Archimedean vector lattice

Characterization of self-majorizing elements in Archimedean vector lattices

Characterization of self-majorizing elements in Archimedean vector lattices

Atomicity of Boolean algebras and vector lattices in terms of order convergence

We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with respect to order convergence if and only the vector lattice is complete and atomic. Additionally we provide a direct proof of the fact that uo convergence on an Archimedean vector lattice is induced by a topology if and only if the vector lattice is atomic.

A representation of sup-completion

It was showed by Donner in [Extension of positive operators and Korovkin theorems, Lecture Notes in Mathematics, vol. 904, Springer-Verlag, Berlin-New York, 1982] that every order complete vector lattice X X may be embedded into a cone X s X^s , called the sup-completion of X X . We show that if one represents the universal completion of X X as C ∞ ( K ) C^\infty (K) , then X s X^s is the set of all continuous functions from K K to [ − ∞ , ∞ ] [-\infty ,\infty ] that dominate some element of X X . This provides a functional representation of X s X^s , as well as an easy alternative proof of its existence.

A generalization of Riesz* homomorphisms on order unit spaces

Riesz homomorphisms on vector lattices have been generalized to Riesz* homomorphisms on ordered vector spaces by van Haandel using a condition on sets of finitely many elements. Van Haandel attempted to prove that it suffices to take sets of two elements. We show that this is not true, in general. The description by two elements motivates to introduce mild Riesz* homomorphisms. We investigate their properties on order unit spaces, where the geometry of the dual cone plays a crucial role. Hereby, we mostly focus on the finite-dimensional case.

Homomorphisms of the lattice of slowly oscillating functions on the half-line

We study the space H(SO) of all homomorphisms of the vector lattice of all slowly oscillating functions on the half-line ℍ = [ 0 , ∞ ) . In contrast to the case of homomorphisms of uniformly continuous functions, it is shown that a homomorphism in H(SO) which maps the unit to zero must be zero-homomorphism. Consequently, we show that the space H(SO) without zero-homomorphism is homeomorphic to ℍ x (0, ∞). By describing a neighborhood base of zero-homomorphism, we show that H(SO) is homeomorphic to the space ℍ x (0, ∞) with one point added.

Extremal structure of cones of positive homogeneous polynomials

It was proved by Anthony Wickstead that the cone of positive linear operators between Banach lattices E and F coincides with the strongly closed convex hull of the set of lattice homomorphisms from E to F if and only if the cone of positive elements on E is the weakly closed convex hull of the union of extremal rays of the cone of positive liner functionals on E. This note aims to show that this result extends to a wider class of operators, namely, orthogonally additive homogeneous polynomials acting between vector lattices.

Embedded unbounded order convergent sequences in topologically convergent nets in vector lattices

Embedded unbounded order convergent sequences in topologically convergent nets in vector lattices

Truncated vector lattices: A Maeda-Ogasawara type representation

Let L be a truncated Archimedean vector lattice whose truncation is denoted by *. In a recent paper, we proved that there exists a locally compact Hausdorff space X such that L is a lattice isomorphic with a truncated vector lattice of functions in C ∞(X) whose truncation is provided by meet with some characteristic function on X. This representation, no matter how interesting it is, has a major drawback, namely, C ∞(X) need not be a vector lattice, unless X is extremally disconnected. The main purpose of this paper is to remedy this shortcoming by proving that, indeed, an extremally disconnected locally compact space X can be found such that L can be seen as an order dense vector sublattice of C ∞(X) whose truncation is provided, again, by meet with some characteristic function on X.

Order boundedness and order continuity properties of positive operator semigroups

Relatively uniformly continuous (ruc) semigroups were recently introduced and studied by Kandic, Kramar-Fijavž, and the second-named author, in order to make the theory of one-parameter operator semigroups available in the setting of vector lattices, where no norm is present in general. In this article, we return to the more standard Banach lattice setting – where both ruc semigroups and C 0 -semigroups are well-defined concepts – and compare both notions. We show that the ruc semigroups are precisely those positive C 0 -semigroups whose orbits are order bounded for small times. We then relate this result to three different topics: (i) equality of the spectral and the growth bound for positive C 0 -semigroups; (ii) a uniform order boundedness principle which holds for all operator families between Banach lattices; and (iii) a description of unbounded order convergence in terms of almost everywhere convergence for nets which have an uncountable index set containing a co-final sequence.

On the diagonal of Riesz operators on Banach lattices

This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete vector lattice E. We prove that the class of regular operators for which the diagonal coincides with the atomic diagonal is always a band in , which contains the band of abstract integral operators. If E is also a Banach lattice, then contains positive Riesz and positive AM-compact operators.