Compact groups containing dense pseudocompact subgroups without non-trivial convergent sequences

Let G be compact abelian group such that w(C(G))=w(Cω(G)). We prove that if

|C(G)|?m(G/C(G)),     

Some remarks on a paper by Liu and van Rooij

Complementing the work of T.-S. Liu and A.C.M. van Rooij we show that the existence of non-zero translation invariant operators between certain function spaces on a locally compact group implies its amenability.     

Pontryagin duality for topological Abelian groups

A topological Abelian group G is Pontryagin reflexive, or P-reflexive for short, if the natural homomorphism of G to its bidual group is a topological isomorphism. We look at the question, set by Kaplan in 1948, of characterizing the topological Abelian groups that are P-reflexive. Thus, we find some conditions on an arbitrary group G that are equivalent to the P-reflexivity of G and give an example that corrects a wrong statement appearing in previously existent characterizations of P-reflexive groups. Received: 10 February 2000 / Published online: 17 May 2001     

On characterized subgroups of compact abelian groups

Let X be a compact abelian group. A subgroup H of X   is called characterized if there exists a sequence u=(_un)$\mathbf{u}=(u_n)$ of characters of X   such that H=_su(X)$H=s_{\mathbf{u}}(X)$, where _su(X):={x∈X:(_un,x)→0 in T}$s_{\mathbf{u}}(X):=\{x\in X:(u_n,x)\to 0\text{in}\mathbb{T}\}$. Every characterized subgroup is an _Fσδ$F_{\sigma \delta }$-subgroup of X  . We show that every _Gδ$G_{\delta }$-subgroup of X is characterized. On the other hand, X   has non-characterized _Fσ$F_{\sigma }$-subgroups.     

Sequences and filters of characters characterizing subgroups of compact Abelian groups

Let H be a countable subgroup of the metrizable compact Abelian group G and a (not necessarily continuous) character of H. Then there exists a sequence of (continuous) characters of G such that limn→∞χn(α)=f(α) for all α∈H and does not converge whenever α∈G?H. If one drops the countability and metrizability requirement one can obtain similar results by using filters of characters instead of sequences. Furthermore the introduced methods allow to answer questions of Dikranjan et al.     

On subgroups of minimal topological groups

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U₁ is the Urysohn universal metric space of diameter 1, the group Iso(U₁) of all self-isometries of U₁ is Roelcke-precompact, topologically simple and minimal. We also show that every topological group is a subgroup of a minimal topologically simple Roelcke-precompact group of the form Iso(M), where M is an appropriate non-separable version of the Urysohn space.     

On the space of subgroups of a compact group I

A theory of the family of closed subgroups of a compact topological group is developed, using the topological notion of a hyperspace. Basic properties of this “space of subgroups” are explored.     

On the space of subgroups of a compact group II

Let G be a compact group. Let S(G), C(G), N(G) be the spaces of closed subgroups, cosets of closed subgroups, normal closed subgroups (respectively) of G, with the Vietoris topology.Then: (1) S(G) and C(G) are never connected; (2) N(G) is always totally disconnected; (3) C(G) is totally disconnected if and only if G is totally disconnected; and (4) S(G) is totally disconnected if and only if G/Z(G) is totally disconnected.Further: for totally disconnected G (equivalently, profinite G) (5) S(G), C(G) and N(G) are κ-metrisable; (6) S(G), C(G) and N(G) are Dugundji compact if G has small weight; and (7) consequences for field extensions are derived.     

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.     

Pontryagin spaces of entire functions I

We give a generalization of L.de Branges theory of Hilbert spaces of entire functions to the Pontryagin space setting. The aim of this-first-part is to provide some basic results and to investigate subspaces of Pontryagin spaces of entire functions. Our method makes strong use of L.de Branges's results and of the extension theory of symmetric operators as developed by M.G.Krein.     

Quasi-convex sequences in the circle and the 3-adic integers

In this paper, we present families of quasi-convex sequences converging to zero in the circle group T, and the group J₃ of 3-adic integers. These sequences are determined by increasing sequences of integers. For an increasing sequence , put gn=an+1−an. We prove that

(a)

the set {0}∪{±³−(an+1)|n∈N} is quasi-convex in T if and only if a₀>0 and gn>1 for every n∈N;

(b)

the set {0}∪{±an³|n∈N} is quasi-convex in the group J₃ of 3-adic integers if and only if gn>1 for every n∈N.

Moreover, we solve an open problem from [D. Dikranjan, L. de Leo, Countably infinite quasi-convex sets in some locally compact abelian groups, Topology Appl. 157 (8) (2010) 1347-1356] providing a complete characterization of the sequences such that {0}∪{±²−(an+1)|n∈N} is quasi-convex in T. Using this result, we also obtain a characterization of the sequences such that the set {0}∪{±²−(an+1)|n∈N} is quasi-convex in R.     

Pontryagin spaces of entire functions II

We continue the study of a generalization of L. de Branges's theory of Hilbert spaces of entire functions to the Pontryagin space setting. In this-second-part we investigate isometric embeddings of spaces of entire functions into spacesL ² () understood in a distributional sense and consider Weyl coefficients of matrix chains. The main task is to give a proof of an indefinite version of the inverse spectral theorem for Nevanlinna functions. Our methods use the theory developed by L. de Branges and the theory of extensions of symmetric operators of M.G.Krein.     

On Some Properties of Permutation Tableaux

We consider the relations between various permutation statistics and properties of permutation tableaux. We answer some of the open problems of Steingrímsson and Williams [8], in particular, on the distribution of the bistatistic of numbers of rows and essential ones in permutation tableaux. We also consider and enumerate sets of permutation tableaux related to some pattern restrictions on permutations. Research supported in part by the NSA Young Investigator Grant H98230-06-1-0037.     

Some identities involving the binomial sequences

Using the exponential generating function and the Bell polynomials, we obtain several new identities for the binomial sequences. As applications, some interesting identities are established for the Abel polynomials, exponential polynomials and factorial powers.     

Processes of flats induced by higher dimensional processes III

Stationary processes of k-flats in ^d can be thought of as point processes on the Grassmannian _k ^d of k-dimensional subspaces of ^d . If such a process is sampled by a (d–k+ j)-dimensional space F, it induces a process of j-flats in F. In this work we will investigate the possibility of determining the original k-process from knowledge of the intensity measures of the induced j-processes. We will see that this is impossible precisely when 1<k<d–1 and j=0,...,2[r/2]–1, where r is the rank of the manifold _k ^d . We will show how the problem is equivalent to the study of the kernel of various integral transforms, these will then be investigated using harmonic analysis on Grassmannian manifolds.The research of the first and third authors was supported in part by NSF grants DMS-9207019 and DMS-9304284. The research of the second author was supported in part by NFR contract number R-RA 4873-306 and the Swedish Academy of Sciences.     

On commutators of equivariant homeomorphisms

The long-standing problem of the perfectness of the compactly supported equivariant homeomorphism group on a G-manifold (with one orbit type) is solved in the affirmative. The proof is based on an argument different than that for the case of diffeomorphisms. The theorem is a starting point for computing H₁(HG(M)) for more complicated G-manifolds.     

Some New Results on the Chu Duality of Discrete Groups

This paper deals mainly with the Chu duality of discrete groups. Among other results, we give sufficient conditions for an FC group to satisfy Chu duality and characterize when the Chu quasi-dual and the Takahashi quasi-dual of a group G coincide. As a consequence, it follows that when G is a weak sum of a family of finite simple groups, if the exponent of the groups in the family is bounded then G satisfies Chu duality; on the other hand, if the exponent of the group goes to infinity, then the Chu quasi-dual of G coincides with its Takahashi quasi-dual. We also present examples of discrete groups whose Chu quasi-duals are not locally compact and examples of discrete Chu reflexive groups which contain non-trivial sequences converging in the Bohr topology of the groups. Our results systematize some previous work and answer some open questions on the subject [2, 16, 3]. The first named author acknowledges partial financial support by the Spanish Ministry of Science (including FEDER funds), grant MTM2004-07665-C02-01; and the Generalitat Valenciana, grant GV04B-019.     

On pure subgroups of locally compact abelian groups

In this note, we construct an example of a locally compact abelian group G = C × D (where C is a compact group and D is a discrete group) and a closed pure subgroup of G having nonpure annihilator in the Pontrjagin dual $\hat{G}$, answering a question raised by Hartman and Hulanicki. A simple proof of the following result is given: Suppose ${\frak K}$ is a class of locally compact abelian groups such that $G \in {\frak K}$ implies that $\hat{G} \in {\frak K}$ and nG is closed in G for each positive integer n. If H is a closed subgroup of a group $G \in {\frak K}$, then H is topologically pure in G exactly if the annihilator of H is topologically pure in $\hat{G}$. This result extends a theorem of Hartman and Hulanicki.Received: 4 April 2002     

Nowhere dense set mappings on the generalized linear continua

A set is free for a set mapping F:XP(X) provided xF(y) for any distinct x, y in A. If F maps the reals R into nowhere dense subsets of R, then Bagemihl proved that there is an everywhere dense free set for F, and assuming CH Hechler showed that such an F does not always admit an uncountable free set. In this paper, we show that Bagemihl's theorem cannot be generalized to the generalized linear continua C for arbitrarily large ordinal , and under GCH we extend Hechler's theorem to C for every .