Compact Metric Abelian Group Research Articles

Ergodic theorem for nonstationary random walks on compact abelian groups

We consider a nonstationary random walk on a compact metrizable abelian group. Under a classical strict aperiodicity assumption we establish a weak-* convergence to the Haar measure, Ergodic Theorem and Large Deviation Type Estimate.

D-independent topological groups

A topological group G with |G|>1 is called d-independent if for every subgroup S of G with |S|<2ω, one can find a countable dense subgroup H of G such that S∩H={e}. Therefore, d-independent groups are separable and have cardinality at least 2ω. Our main result is a purely algebraic characterization of d-independence in the class of compact metrizable abelian groups. We prove that a compact metrizable abelian group G with |G|>1 is d-independent if and only if for every integer m≥1, either |mG|=2ω or |mG|=1. This characterization implies that a compact metrizable abelian group is d-independent if and only if it is maximally fragmentable [Comfort and Dikranjan (2014) [4]] iff G an M-group as defined by Dikranjan and Shakhmatov (2016) in [7].Also we present a characterization of separable metrizable d-independent abelian groups and show that products of separable topological groups can often be d-independent, even if the factors fail to be d-independent.

Polish LCA groups are strongly characterizable

Positively answering a question of Gabriyelyan [21, Problem 2.13], we prove that for every second countable locally compact abelian group G and every continuous monomorphism s:G→K into a compact metrizable abelian group K, there exists a sequence of characters (un) of K characterizing the subgroup s(G), i.e., s(G) coincides with the subset of those elements k∈K such that (un(k)) is a null-sequence in the circle group T.

Characterizable groups: Some results and open questions

Let X be an Abelian topological group and X∧ its dual group. A subgroup H of X is called characterized if there is a sequence {un} in X∧ such that H={x∈X:(un,x)→1}. A Polish Abelian group G is called characterizable if there is a continuous monomorphism p from G into a compact metrizable Abelian group X with dense image such that p(G) is a characterized subgroup of X. Every characterizable group is locally quasi-convex. We prove that every second countable locally compact Abelian group X is characterizable. Thus, every second countable locally compact Abelian group is the dual group of a complete countable maximally almost periodic group. It is shown that each characterizable Abelian group of finite exponent is locally compact. Analogously to the Abelian case, we define characterized subgroups of non-Abelian compact metrizable groups and non-Abelian characterizable groups. Using the ℓp-sum of metric groups with two-sided invariant metrics, it is proved that every characterized subgroup admits a Polish group topology.

VECTOR MEASURES OF BOUNDED SEMIVARIATION AND ASSOCIATED CONVOLUTION OPERATORS

AbstractLet G be a compact metrizable abelian group, and let X be a Banach space. We characterize convolution operators associated with a regular Borel X-valued measure of bounded semivariation that are compact (resp; weakly compact) from L1(G), the space of integrable functions on G into L1(G) X, the injective tensor product of L1(G) and X. Along the way we prove a Fourier Convergence theorem for vector measures of relatively compact range that are absolutely continuous with respect to the Haar measure.

On T-sequences and characterized subgroups

Let X be a compact metrizable abelian group and u = { u n } be a sequence in its dual group X ∧ . Set s u ( X ) = { x : ( u n , x ) → 1 } and T 0 H = { ( z n ) ∈ T ∞ : z n → 1 } . Let G be a subgroup of X. We prove that G = s u ( X ) for some u iff it can be represented as some dually closed subgroup G u of Cl X G × T 0 H . In particular, s u ( X ) is polishable. Let u = { u n } be a T-sequence. Denote by ( X ˆ , u ) the group X ∧ equipped with the finest group topology in which u n → 0 . It is proved that ( X ˆ , u ) ∧ = G u and n ( X ˆ , u ) = s u ( X ) ⊥ . We also prove that the group generated by a Kronecker set cannot be characterized.

Strong characterizing sequences for subgroups of compact groups

In [A. Biró, V.T. Sós, Strong characterizing sequences in simultaneous Diophantine approximation, J. Number Theory 99 (2003) 405–414] we proved that if Γ is a subgroup of the torus R / Z generated by finitely many independent irrationals, then there is an infinite subset A ⊆ Z which characterizes Γ in the sense that for γ ∈ R / Z we have ∑ a ∈ A ‖ a γ ‖ < ∞ if and only if γ ∈ Γ . Here we consider a general compact metrizable Abelian group G instead of R / Z , and we characterize its finitely generated free subgroups Γ by subsets A ⊆ G * , where G * is the Pontriagin dual of G. For this case we prove stronger forms of the analogue of the theorem of the above mentioned work, and we find necessary and sufficient conditions for a kind of strengthening of this statement to be true.

Characterizing subgroups of compact abelian groups

We prove that every countable subgroup of a compact metrizable abelian group has a characterizing set. As an application, we answer several questions on maximally almost periodic (MAP) groups and give a characterization of the class of (necessarily MAP) abelian topological groups whose Bohr topology has countable pseudocharacter.

On the near differentiability property of Banach spaces

Let μ be a scalar measure of bounded variation on a compact metrizable abelian group G. Suppose that μ has the property that for any measure σ whose Fourier–Stieltjes transform σ ˆ vanishes at ∞, the measure μ * σ has Radon–Nikodým derivative with respect to λ, the Haar measure on G. Then L. Pigno and S. Saeki showed that μ itself has Radon–Nikodým derivative. Such property is not shared by vector measures in general. We say that a Banach space X has the near differentiability property if every X-valued measure of bounded variation shares the above property. We prove that Banach spaces with the Radon–Nikodým property have the near differentiability property, while Banach spaces with the near differentiability property enjoy the near Radon–Nikodým property. We also show that the Banach spaces L 1 [ 0 , 1 ] and L 1 / H 0 1 have the near differentiability property. Lastly, we show that Banach spaces with the near differentiability property have type II- Λ-Radon–Nikodým property, whenever Λ is a Riesz subset of type 0 of G ˆ .

Complete continuity properties of Banach spaces associated with subsets of a discrete abelian group

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. We introduce and study the type I-, II-, and III-Λ-complete continuity property of Banach spaces, where Λ is a subset of the dual group of a compact metrizable abelian group G.

Ergodic Automorphisms of $T^∞$ Are Bernoulli Transformations

The purpose of this paper is to prove the fact stated by the title. After Ornstein and Friedman [5], [1], several authors studied the Bernoulli properties of various transformations. Especially Katznelson [3] proved that every ergodic automorphism of a finite-dimensional torus is a Bernoulli transformation extending the results by Sinai-Ornstein-Friedman [9], [1]. The present result is on the way towards the conjecture that every ergodic automorphism of a compact metrizable abelian group is a Bernoulli transformation. Let X be a compact metrizable group and /^ be its normalized Haar measure. Then (X, ft) is a Lebesgue space (cf. [10]). Let a be an (group) automorphism of X, then cr is an invertible measure-preserving transformation of (X, fji). Our problem is concerned with measure-theoretic properties of a. We call a a Bernoulli transformation if there exists a measurable partition f of X such that {o-^}_00<??/<00 are independent and V o-f=£ the partition of X into individual points. We assume X—T°° an infinite-dimensional torus i.e. X is a compact metrizable abelian group which is connected, locally connected and infinite-dimensional. Further we assume naturally that the automorphism a is ergodic i.e. any cr-invariant measurable set has measure 0 or 1. Our result is the following