LCA Groups Research Articles

Locally compact abelian groups and the variety of topological groups generated by the reals

An LCA group

Mapping properties of characters of LCA groups

Mapping properties of characters of LCA groups

Sufficiency Classes of LCA Groups

Sufficiency Classes of LCA Groups

Locally compact abelian groups with trivial multiplier group

LCA groups with trivial multiplier group are characterized. It is found that every LCA group with nontrivial multiplier group has an “irreducible” quotient with the same property, and the irreducible groups are described.

Can an 𝐿𝐶𝐴 group be anti-self-dual?

We say that a topological group G G is anti-self-dual if there are no nontrivial continuous homomorphisms from G G into the character group G ^ \hat G or from G ^ \hat G into G G . We show that no nontrivial LCA group can be anti-self-dual.

Can an LCA Group be Anti-Self-Dual?

Can an LCA Group be Anti-Self-Dual?

Well-Known LCA Groups Characterized by their Closed Subgroups

Well-Known LCA Groups Characterized by their Closed Subgroups

Structure of self dual torsion free metric LCA groups

Structure of self dual torsion free metric LCA groups

Mutual subordination of multivariate stationary processes over any locally compact abelian group

Our purpose is to extend Kolmogorov's theorem [5, Th. 10] on mutual subordination for univariate weakly stationary stochastic processes over the (discrete) group of integers to multivariate processes over any (Hausdorff) locally compact abelian (lca) group. This extension is given in Theorems (1.12) and (3.4) below. We shall lean heavily on the joint paper [10] on the decomposition of matricial measures, to which the present paper may be regarded as a sequel.

Uniform distribution in locally compact Abelian groups

Introduction. The notion of a uniformly distributed sequence on a compact group was introduced and studied by Eckmann in [1], and has since been the object of much research. Recently, this notion was extended to locally compact groups by Rubel in [9]. Our principal interest here is in the following two questions. First, when does a locally compact Abelian group G possess a uniformly distributed sequence? Second, when is there an associated compact group Gand a continuous homomorphism (p: G -> G-, such that T(G) is dense in G-, with the property that {g,} is uniformly distributed in G if and only if {p(g,)} is uniformly distributed in G? We call such a group Ga D-compactification of G. We obtain solutions to these and related problems. In this introduction, we present various definitions and preliminary results and describe briefly the main results of the paper. In ?1, we prove some structural lemmas. In ?2, we consider the existence of uniformly distributed sequences on G. Finally, in ?3, we consider the existence of D-compactifications of G. By G, we will denote a locally compact Abelian group. We will, in general, assume that all groups are locally compact Abelian (abbreviated LCA) groups. We do not require that our groups be first-countable. Indeed, even when we start with first-countable groups G, we will be forced to consider certain compactifications which will rarely be first-countable. Throughout this paper, we shall assumer the continuum hypothesis, denoting by c the cardinal number of the continuum. If A is a set, we denote the cardinal number of A by card A. Characters of G are supposed to be continuous characters unless otherwise stated. We will denote the trivial character by 1. The dual group of G will be denoted by r or by G^. If G is an LCA group, we will call a compact group H a compactification of G if there is a continuous homomorphism from G onto a dense subgroup of H. For example, G, the Bohr compactification of G, (defined as the dual of rd, where

Density and representation theorems for multipliers of type (p, q)

Let G be a locally compact Abelian Hausdorff group (abbreviated LCA group); let X be its character group and dx, dx be the elements of the normalised Haar measures on G and X respectively. If 1 < p, q < ∞, and Lp(G) and Lq(G) are the usual Lebesgue spaces, of index p and q respectively, with respect to dx, a multiplier of type (p, q) is defined as a bounded linear operator T from Lp(G) to Lq(G) which commutes with translations, i.e. τxT = Tτx for all x ∈ G, where τxf(y) = f(x+y). The space of multipliers of type (p, q) will be denoted by Lqp. Already, much attention has been devoted to this important class of operators (see, for example, [3], [4], [7]).