Locally minimal topological groups 2

We continue in this paper the study of locally minimal groups started in Außenhofer et al. (2010) [4]. The minimality criterion for dense subgroups of compact groups is extended to local minimality. Using this criterion we characterize the compact abelian groups containing dense countable locally minimal subgroups, as well as those containing dense locally minimal subgroups of countable free-rank. We also characterize the compact abelian groups whose torsion part is dense and locally minimal. We call a topological group G almost minimal if it has a closed, minimal normal subgroup N such that the quotient group G/N is uniformly free from small subgroups. The class of almost minimal groups includes all locally compact groups, and is contained in the class of locally minimal groups. On the other hand, we provide examples of countable precompact metrizable locally minimal groups which are not almost minimal. Some other significant properties of this new class are obtained.     

The interior of the Šilov boundary of M(G) is trivial

Let G be a non-discrete locally compact abelian group, and let M(G) be the convolution algebra of regular bounded Borel measures on G. Let Γ denote the dual group of G. Then the interior of the ?ilov boundary of M(G) is exactly Γ. The proof uses generalized Riesz products for the compact metrizable case and standard liftings from that case.     

Reflexivity of prodiscrete topological groups

We study the duality properties of two rather different classes of subgroups of direct products of discrete groups (protodiscrete groups): P-groups, i.e., topological groups such that countable intersections of its open subsets are open, and protodiscrete groups of countable pseudocharacter (topological groups in which the identity is the intersection of countably many open sets). It was recently shown by the same authors that the direct product Π of an arbitrary family of discrete Abelian groups becomes reflexive when endowed with the ω-box topology. This was the first example of a non-discrete reflexive P-group. Here we present a considerable generalization of this theorem and show that every product of feathered (equivalently, almost metrizable) Abelian groups equipped with the P-modified topology is reflexive. In particular, every locally compact Abelian group with the P-modified topology is reflexive. We also examine the reflexivity of dense subgroups of products Π with the P-modified topology and obtain the first examples of non-complete reflexive P-groups. We find as well that the better behaved class of prodiscrete groups (complete protodiscrete groups) of countable pseudocharacter contains non-reflexive members—any uncountable bounded torsion Abelian group G of cardinality ω² supports a topology τ such that (G,τ) is a non-reflexive prodiscrete group of countable pseudocharacter.     

Strong Reflexivity of Abelian Groups

A reflexive topological group G is called strongly reflexive if each closed subgroup and each Hausdorff quotient of the group G and of its dual group is reflexive. In this paper we establish an adequate concept of strong reflexivity for convergence groups. We prove that complete metrizable nuclear groups and products of countably many locally compact topological groups are BB-strongly reflexive.     

Essential subgroups of topological groups

We study the class Wof Hausdorff topological groups Gfor which the following two cardinal invariants coincide

ES(G)=min{|H|:H≤ Gdense and essential}

TD(G)=min{|H|:H≤ Gtotally dense}

We prove that W contains the following classes:locally compact abelian groups, compact connected groups, countably compact totally discon¬nected abelian groups, topologically simple groups, locally compact Abelian groups when endowed with their Bohr topology, totally minimal abelian groups and free Abelian topological groups. For all these classes we are also able to giv ean explicit computation of the common value of ESand TD.     

Quasi‐convexly dense and suitable sets in the arc component of a compact group

Let G be an abelian topological group. The symbol $\widehat{G}Let G be an abelian topological group. The symbol $\widehat{G}$ denotes the group of all continuous characters $\chi :G\rightarrow {\mathbb T}$ endowed with the compact open topology. A subset E of G is said to be qc‐dense in G provided that χ(E)?φ([? 1/4, 1/4]) holds only for the trivial character $\chi \in \widehat{G}$, where $\varphi : {\mathbb R}\rightarrow {\mathbb T}={\mathbb R}/{\mathbb Z}$ is the canonical homomorphism. A super‐sequence is a non‐empty compact Hausdorff space S with at most one non‐isolated point (to which S converges). We prove that an infinite compact abelian group G is connected if and only if its arc component G_a contains a super‐sequence converging to 0 that is qc‐dense in G. This gives as a corollary a recent theorem of Außenhofer: For a connected locally compact abelian group G, the restriction homomorphism $r:\widehat{G}\rightarrow \widehat{G}_a$ defined by $r(\chi )=\chi \upharpoonright _{G_a}$ for $\chi \in \widehat{G}$, is a topological isomorphism. We show that an infinite compact group G is connected if and only if its arc component G_a contains a super‐sequence converging to the identity that is qc‐dense in G and generates a dense subgroup of G. We also offer a short alternative proof of the result of Hofmann and Morris on the existence of suitable sets of minimal size in the arc component of a compact connected group.     

The structure of abelian pro-Lie groups

A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups.Mathematics Subject Classification (1991): 22B, 22E     

Counting maximal topologies on countable groups and rings

Two non-discrete Hausdorff group topologies τ₁, τ₂ on a group G are called transversal if the least upper bound τ₁∨τ₂ of τ₁ and τ₂ is the discrete topology. We show that a countable group G admitting non-discrete Hausdorff group topologies admits c² pairwise transversal complete group topologies on G (so c² maximal group topologies). Moreover, every abelian group G admits ²|G|² pairwise transversal complete group topologies.     

Actions of totally disconnected groups and equivariant singular homology

We study equivariant singular homology in the case of actions of totally disconnected locally compact groups on topological spaces. Theorem A says that if G is a totally disconnected locally compact group and X is a G-space, then any short exact sequence of covariant coefficient systems for G induces a long exact sequence of corresponding equivariant singular homology groups of the G-space X. In particular we consider the case where G is a totally disconnected compact group, i.e., a profinite group, and G acts freely on X. Of special interest is the case where G is a p-adic group, p a prime. The conjecture that no p-adic group, p a prime, can act effectively on a connected topological manifold, is namely known to be equivalent to the famous Hilbert-Smith conjecture. The Hilbert-Smith conjecture is the statement that, if a locally compact group G acts effectively on a connected topological manifold M, then G is a Lie group.     

Characterizing sequences for precompact group topologies

Motivated from [31], call a precompact group topology τ on an abelian group G ss-precompact (abbreviated from single sequence precompact  ) if there is a sequence u=(_un)$\mathbf{u}=(u_n)$ in G such that τ is the finest precompact group topology on G   making u=(_un)$\mathbf{u}=(u_n)$ converge to zero. It is proved that a metrizable precompact abelian group (G,τ)$(G,\tau )$ is ss-precompact iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ   and the groups (G,τ)$(G,\tau )$ and (G,η)$(G,\eta )$ have the same Pontryagin dual groups (in other words, (G,τ)$(G,\tau )$ is not a Mackey group in the class of maximally almost periodic groups).     

First countability, tightness, and other cardinal invariants in remainders of topological groups

We consider the following natural questions: when a topological group G has a first countable remainder, when G has a remainder of countable tightness? This leads to some further questions on the properties of remainders of topological groups. Let G be a topological group. The following facts are established. 1. If Gω has a first countable remainder, then either G is metrizable, or G is locally compact. 2. If G has a countable network and a first countable remainder, then either G is separable and metrizable, or G is σ-compact. 3. Under (MA+¬CH) every topological group with a countable network and a first countable remainder is separable and metrizable. Some new open problems are formulated.     

Both compact and sequentially compact sets in abelian topological group

We show that every abelian topological group contains many interesting sets which are both compact and sequentially compact. Then we can deduce some useful facts, e.g.,

(1)

if G is a Hausdorff abelian topological group and μ:N²→G is countably additive, then the range μ(N²)={μ(A):A⊆N} is compact metrizable;

(2)

if X is a Hausdorff locally convex space and {xj}⊂X, then F={_∑j∈Δxj:Δ⊂N, Δ is finite} is relatively compact in (X,weak) if and only if F is relatively compact in X, and if and only if F is relatively compact in (X,F(M)) where F(M) is the Dierolf topology which is the strongest 〈X,X^′〉-polar topology having the same subseries convergent series as the weak topology.

     

On coverings in the lattice of all group topologies of arbitrary Abelian groups

The remainder of the completion of a topological abelian group (G, τ₀) contains a nonzero element of prime order if and only if G admits a Hausdorff group topology τ₁ that precedes the given topology and is such that (G, τ₀) has no base of closed zero neighborhoods in (G, τ₁).     

CONTINUOUS FRAME WAVELETS

Let π be a unitary representation of a locally compact topological group G on a separable Hilbert space H.A vector ψ ∈ H is called a continuous frame wavelet if there exist A,B ＞ 0 such that A‖φ‖2 ≤∫G ...     

Continuity in topological groups

We prove that, when G is a group equipped with a Baire and metrizable topology, if there is a second category dense subset S of G such that the right translations ρs and ρs⁻¹ are continuous for all s∈S and each left translation λs, s∈G, is almost-continuous (defined below) on a residual subset of G, then G is a topological group. Among other consequences, this yields that when G is a second countable locally compact right topological group, its topological centre is a topological group.     

Metrizability of spaces of holomorphic functions

In this paper we prove that if U is an open subset of a metrizable locally convex space E of infinite dimension, the space H(U) of all holomorphic functions on U, endowed with the Nachbin-Coeuré topology τδ, is not metrizable. Our result can be applied to get that, for all usual topologies, H(U) is metrizable if and only if E has finite dimension.