Commutative Topological Groups Research Articles

A generalization of Gajda's equation on commutative topological groups

A generalization of Gajda's equation on commutative topological groups

Using R - Multiplier in Optimization

Multiplier as a Method of Optimization. The paper is part of the start of the investigation to use ℝ-Multiplier as an optimizing mapping. An ℝ-multiplier is a mapping, ρ: G X G → ℝ , where, in this case, G is a commutative topological group, such that ρ(xy,z)ρ(x,y) =ρ(x,yz)ρ(y,z) (1) ρ(x,e) =ρ(e,x ) = 1. (2) The motivation to embark on the study is due to the following theorem THEOREM 1. Find x ϵ M such that the continuous mapping, σ: M → ℝ assumes a maximum or a minimum at some point of M, where M is a subgroup of a commutative topological group, G. This theorem is due to Kreyszig(1978) with some adaptations This paper proves, using a constructive method of proof, that an ℝ- multiplier is a continuous mapping; thus making an ℝ-multiplier a mapping which assumes a minimum or maximum at some point.

On a certain application of the theory of characters for commutative groups

We study a certain application of the theory of characters for commutative compact topological groups and weakly linearly compact groups.

Asymptotically exhaustive functions in Vitali spaces

We introduce asymptotically exhaustive functions defined on Vitali Spaces with values in a Hausdorff commutative topological group and we prove for them some classical convergence theorems.

Cafiero and Brooks–Jewett theorems for Vitali spaces

We introduce a common generalization of Boolean rings and lattice ordered groups called Vitali spaces and we give a version of Cafiero and Brooks–Jewett convergence Theorems for additive functions defined in a Vitali space with values in a Hausdorff commutative topological group.

On the orbits of G-closure points of ultimately nonexpansive mappings

Let be a closed subset of a Banach space and an ultimately nonexpansive commutative semigroup of continuous selfmappings. If the -closure of is nonempty, then the closure of the orbit of any -closure point is a commutative topological group.

Moment functions on polynomial hypergroups

The study of moment functions on commutative topological groups leads to the study of systems of functional equations. Systems of functional equations characterizing moment functions and sequences of moment functions are closely related to exponential functions and additive functions. Here we describe moment functions and generalized moment functions on polynomial hypergroups.

On homomorphism spaces of metrizable groups

For two not necessarily commutative topological groups G and K, let H(G,K) denote the space of all continuous homomorphisms from G to K with the compact-open topology. We prove that if G is metrizable and K is compact then H(G,K) is a k-space. As a consequence we obtain that if D is a dense subgroup of G then H(D,K) is homeomorphic to H(G,K) , and if G is separable h-complete, then the natural map G→ C( H(G,K),K) is open onto its image.

Convergence theorems for topological group valued measures on effect algebras

In this paper we study the validity of several convergence theorems for measures defined on an effect algebra and taking values in a Hausdorff commutative topological group. We establish the Brooks-Jewett theorem and the Nikodým convergence theorem, giving as a corollary a result, due to Aarnes, about the convergence of a sequence of normal linear functionals on a von Neumann algebra. We prove two new convergence theorems concerning completely additive and τ-smooth measures, and we obtain also a convergence theorem for regular measures.

Duality theory for convergence groups

The well-known Pontryagin Duality Theorem states that a locally compact, commutative topological group is isomorphic to its second character group, i.e., the character group of its character group. Here the character group carries the compact-open topology. There are various papers dealing with generalizations of the theorem to not necessarily locally compact commutative groups. In 1971 we suggested to replace the compact-open topology in the general case by the continuous convergence structure. This structure coincides with the compact-open topology if the group is locally compact and gives at least better categorical properties in the so-called c-duality theory. We just mention the fact that the natural mapping from a convergence group (always assumed to be commutative) into its second c-character group is always continuous. In recent years this approach has attracted again attention. In this paper we study the behaviour of the c-duality under the usual topological constructions. We show that the c-character group of a product of convergence groups is isomorphic the the coproduct (in the category of convergence groups) of the c-character groups and that the c-character group of a coproduct is isomorphic the the product of the c-character groups. So products and coproducts of c-reflexive convergence groups are c-reflexive again. Also the character group of a quotient group is isomorphic to its annihilator group. As it is well known, subgroups are very difficult to handle, since its characters can not always be extended to the whole group. We give sufficient conditions to guarantee that the c-character group of a subgroup is isomorphic to a quotient of the c-character group of the whole group and also that a subgroup of a c-reflexive group is c-reflexive. In the last section we handle topological groups. We show that the c-character group of a topological group is locally compact and so the second c-character group is also topological. We show that the natural mapping from the group into its second character group is an embedding if and only if the group is locally quasi-convex, a notion introduced by Banaszczyk in 1991.

First Alexandroff Decomposition Theorem for Topological Lattice Group Valued Measures

Let (X, F) be an Alexandroff space, let A(F) be the Boolean subalgebra of 2X generated by F, let G be a Hausdorff commutative topological lattice group and let rbaF(A(F), G) denote the set of all order bounded F-inner regular finitely additive set functions from A(F) into G. Using some special properties of the elements of rbaF(A(F), G), we extend to this setting the first decomposition theorem of Alexandroff.

On uniform boundedness properties in exhausting additive set function spaces

SynopsisValdivia (1978) introduced the class of suprabarrelled spaces, and (1979) deduced some uniform boundedness properties for scalar valued exhausting additive set functions on a σ-algebra from the suprabarrelledness of certain spaces. In this paper, it is shown that those uniform boundedness properties hold for G-valued exhausting additive set functions, G being a commutative topological group, on a larger class of Boolean algebras. Such properties are proved in Valdivia (1979) by means of duality theory arguments and ‘sliding hump’ methods, whereas here they are derived from the Baire category theorem. This generalization enables us to find a wide class of compact topological spaces K such that the subspaces of C(K) which satisfy a mild property are suprabarrelled.

The structure of lattice-ordered commutative topological groups of compact origin

It is shown that every lattice-ordered commutative separable topological group of compact origin can be obtained from a finite number of its linearly ordered subgroups, each of which is isomorphic either to the additive group of real numbers with the natural topology and the usual order or to a subgroup of the additive group of real numbers with the discrete topology and the usual order, admitting a finite system of linearly independent generators, by forming in turn the direct and the lexicographic products.

Isotopy of Abelian Quasigroups

It is proved that every abelian quasigroup possesses a class of isomorphic principal isotopes which are commutative groups. Also it is indicated how the corresponding result for topological quasigroups can be utilised to obtain results for abelian topological quasigroups from analogous results for commutative topological groups.

Isotopy of abelian quasigroups

It is proved that every abelian quasigroup possesses a class of isomorphic principal isotopes which are commutative groups. Also it is indicated how the corresponding result for topological quasigroups can be utilised to obtain results for abelian topological quasigroups from analogous results for commutative topological groups.

Unique reducibility of subsets of commutative topological groups and semigroups.

Abstract : As the term is used here, a reduction of a set is a direct sum decomposition into indecomposable summands. The main goal is to find conditions under which reductions are literally unique, but weaker sorts of uniqueness (akin to that of the Krull-Schmidt theorem) are also considered. The problem of unique reducibility is a classical one in many contexts, but our approach - particular, its exploitation of the key geometric notion of extreme point in conjunction with combinatorial methods involving the refinement property - appears to be new. Special cases of the main result have been obtained by Isbell in studying factorizations of Banach speaces and by Heller in studying stochastic automata.

-spaces and the closed-graph theorem

The problem considered in this paper is that of finding conditions on a range space such that the closed-graph theorem holds for linear mappings from a class of linear topological spaces. The concept of a -space, which is a result of this investigation, is meaningful for commutative topological groups but we limit our consideration in this paper to linear topological spaces. On restricting ourselves to locally convex linear topological spaces, we see that the notion of a -space is an extension of the powerful idea of a B-complete space.

Inductive limits of increasing-sequences of commutative topological groups. (I)

The abstracts (in two languages) can be found in the pdf file of the article.
 Original author name(s) and title in Russian and Lithuanian:
 И. В. Карклиньш. Индуктивные пределы возрастающих последовательностей коммутативных топологических групп. (I)
 J. Karklinš. Komutatyvinių topologinių grupių didėjančių sekų induktyvinės ribos. (I)