Discrete Abelian Groups Research Articles

Minimal growth function of uniform amenability for discrete abelian groups

Minimal growth function of uniform amenability for discrete abelian groups

Topologically independent sets in precompact groups

It is a simple fact that a subgroup generated by a subset A of an abelian group is the direct sum of the cyclic groups 〈a〉, a∈A if and only if the set A is independent. In [2] the concept of an independent set in an abelian group was extended to a topologically independent set in a topological abelian group (these two notions coincide in discrete abelian groups). It was proved that a topological subgroup generated by a subset A of an abelian topological group is the Tychonoff direct sum of the cyclic topological groups 〈a〉, a∈A if and only if the set A is topologically independent and absolutely Cauchy summable. Further, it was shown, that the assumption of absolute Cauchy summability of A can not be removed in general in this result. In our paper we show that it can be removed in precompact groups.In other words, we prove that if A is a subset of a precompact abelian group, then the topological subgroup generated by A is the Tychonoff direct sum of the topological cyclic subgroups 〈a〉, a∈A if and only if A is topologically independent. We show that precompactness can not be replaced by local compactness in this result.

On Spectral Analysis and Spectral Synthesis in the Space of Tempered Functions on Discrete Abelian Groups

We consider some problems of spectral analysis and spectral synthesis in the topological vector space $${{\mathcal {M}}}(G)$$ of tempered functions on a discrete Abelian group G. It is proved that spectral analysis holds in the space $${{\mathcal {M}}}(G)$$ on every Abelian group G, that is, every nonzero closed linear translation invariant subspace of $${{\mathcal {M}}}(G)$$ contains an exponential. For any finitely generated Abelian group G it is proved, that spectral synthesis holds in $${{\mathcal {M}}}(G)$$ , that is, every closed linear translation invariant subspace $${{\mathscr {H}}}$$ of $${{\mathcal {M}}}(G)$$ coincides with the closed linear span of all exponential monomials belonging to $${{\mathscr {H}}}$$ . For any Abelian group G with infinite torsion free rank it is proved that spectral synthesis fails to hold in the space $${{\mathcal {M}}}(G)$$ .

ON GRADED -ALGEBRAS

We show that every topological grading of a $C^{\ast }$-algebra by a discrete abelian group is implemented by an action of the compact dual group.

Study of texture zeros of fermion mass matrices in minimal extended seesaw mechanism and symmetry realization

In our work, we study the texture zeros of [Formula: see text] in the minimal extended type-I seesaw (MES) with incorporating one extra gauge singlet field “[Formula: see text]”. The MES models deal with the Dirac neutrino mass matrix [Formula: see text], the right-handed Majorana mass matrix [Formula: see text] and the sterile neutrino mass matrix [Formula: see text]. We carry out the mapping of all possible zero textures of [Formula: see text], [Formula: see text] and [Formula: see text] for phenomenologically predictive cases having total eight zeros of [Formula: see text] and [Formula: see text] studied in the literature. With this motivation, we consider (a) [Formula: see text] scheme, (b) [Formula: see text] scheme and (c) [Formula: see text] scheme, where the digits of a pair represent the number of zeros of [Formula: see text] and [Formula: see text], respectively along with the one zero and two zero textures of [Formula: see text]. There are a large number of possibilities of zeros of fermion mass matrices but the implementation of [Formula: see text] transformations reduces it to a very minimum number of basic structures. Interestingly out of four allowed one zero textures of [Formula: see text] without sterile neutrino, only three cases ([Formula: see text], [Formula: see text] and [Formula: see text]) are allowed in MES mechanism for the [Formula: see text] and [Formula: see text] schemes. We find some correlations for different combination of [Formula: see text], [Formula: see text] and [Formula: see text] on enforcement of zeros. We examined all the correlations under the recent neutrino oscillation data and find that only [Formula: see text] survives while both [Formula: see text] and [Formula: see text] are ruled out. Interestingly the one zero textures inherently represent the inverted hierarchy of the mass ordering of light neutrinos. No two zero textures of [Formula: see text] survive in MES although there are a number of allowed structures phenomenologically. The allowed texture zeros are finally realized using a discrete Abelian flavor symmetry group [Formula: see text] with the extension of standard model to include some scalar fields.

Spectral synthesis and residually finite-dimensional algebras

In 2004, a counterexample was given for a 1965 result of R. J. Elliott claiming that discrete spectral synthesis holds on every Abelian group. Since then the investigation of discrete spectral analysis and synthesis has gained traction. Characterizations of the Abelian groups that possess spectral analysis and spectral synthesis, respectively, were published in 2005. A characterization of the varieties on discrete Abelian groups enjoying spectral synthesis is still missing. We present a ring theoretical approach to the issue. In particular, we provide a generalization of the Principal Ideal Theorem on discrete Abelian groups.

A countable free closed non-reflexive subgroup of ℤ^{𝔠}

We prove that the group G = H o m ( Z N , Z ) G=\mathrm {Hom}(\mathbb {Z}^{\mathbb {N}}, \mathbb {Z}) of all homomorphisms from the Baer-Specker group Z N \mathbb {Z}^{\mathbb {N}} to the group Z \mathbb {Z} of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual of G G is discrete. As G G is non-discrete, it is not reflexive. Since G G can be viewed as a closed subgroup of the Tychonoff product Z c \mathbb {Z}^{\mathfrak {c}} of continuum many copies of the integers Z \mathbb {Z} , this provides an example of a group described in the title, thereby resolving a problem by Galindo, Recoder-Núñez and Tkachenko. It follows that an inverse limit of finitely generated (torsion-)free discrete abelian groups need not be reflexive.

Phenomenological study of extended seesaw model for light sterile neutrino

We study the zero textures of the Yukawa matrices in the minimal extended type-I seesaw (MES) model which can give rise to ∼ eV scale sterile neutrinos. In this model, three right handed neutrinos and one extra singlet S are added to generate a light sterile neutrino. The light neutrino mass matrix for the active neutrinos, mν, depends on the Dirac neutrino mass matrix (MD), Majorana neutrino mass matrix (MR) and the mass matrix (MS) coupling the right handed neutrinos and the singlet. The model predicts one of the light neutrino masses to vanish. We systematically investigate the zero textures in MD and observe that maximum five zeros in MD can lead to viable zero textures in mν. For this study we consider four different forms for MR (one diagonal and three off diagonal) and two different forms of (MS) containing one zero. Remarkably we obtain only two allowed forms of mν (meτ = 0 and mττ = 0) having inverted hierarchical mass spectrum. We re-analyze the phenomenological implications of these two allowed textures of mν in the light of recent neutrino oscillation data. In the context of the MES model, we also express the low energy mass matrix, the mass of the sterile neutrino and the active-sterile mixing in terms of the parameters of the allowed Yukawa matrices. The MES model leads to some extra correlations which disallow some of the Yukawa textures obtained earlier, even though they give allowed one-zero forms of mν. We show that the allowed textures in our study can be realized in a simple way in a model based on MES mechanism with a discrete Abelian flavor symmetry group Z8 × Z2.

Jamison sequences in countably infinite discrete Abelian groups

We extend the definition of Jamison sequences in the context of topological abelian groups. Then we study such sequences when the abelian group is discrete and countably infinite. An arithmetical characterization of such sequences is obtained, extending the result of Badea and Grivaux about Jamison sequences of integers. In particular, we prove that the sequence consisting of all elements of the group is a Jamison sequence. In the opposite, a sequence which generates a subgroup of infinite index in the group is never a Jamison sequence.

The ubiquity of Sidon sets that are not I0

We prove that every infinite, discrete abelian group admits a pair of $I_0$ sets whose union is not $I_0$. In particular, this implies that every such group contains a Sidon set that is not $I_{0}$.

The Relationship Between ϵ-Kronecker Sets and Sidon Sets

AbstractA subset E of a discrete abelian group is called ϵ-Kronecker if all E-functions of modulus one can be approximated to within ϵ by characters. E is called a Sidon set if all bounded E-functions can be interpolated by the Fourier transform of measures on the dual group. As ϵ-Kronecker sets with ϵ < 2 possess the same arithmetic properties as Sidon sets, it is natural to ask if they are Sidon. We use the Pisier net characterization of Sidonicity to prove this is true.

On the principal ideal theorem and spectral synthesis on discrete Abelian groups

On the principal ideal theorem and spectral synthesis on discrete Abelian groups

Discrete Curvature and Abelian Groups

AbstractWe study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of “curvature” in discrete spaces. An appealing feature of this discrete version of the so-called Γ2-calculus (of Bakry-Émery) seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specific graphs of interest, particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (à la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs, a result of independent interest.

On the powers of maximal ideals in the measure algebra

In this paper, we describe the powers of maximal ideals in the measure algebra of some locally compact Abelian groups in terms of the derivatives of the Fourier–Laplace transform of compactly supported measures. We show that if the locally compact Abelian group has sufficiently many real characters, then all derivatives of the Fourier–Laplace transform of a measure at some point of its spectrum completely characterize the measure. We also show that the derivatives of the Fourier–Laplace transform of a measure can be used to describe the powers of the maximal ideals corresponding to the points of the spectrum of the measure on discrete Abelian groups with finite torsion-free rank.

Cartan Subalgebras in C*-Algebras of Haus dorff étale Groupoids

The reduced $C^*$-algebra of the interior of the isotropy in any Hausdorff \'etale groupoid $G$ embeds as a $C^*$-subalgebra $M$ of the reduced $C^*$-algebra of $G$. We prove that the set of pure states of $M$ with unique extension is dense, and deduce that any representation of the reduced $C^*$-algebra of $G$ that is injective on $M$ is faithful. We prove that there is a conditional expectation from the reduced $C^*$-algebra of $G$ onto $M$ if and only if the interior of the isotropy in $G$ is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, $M$ is a Cartan subalgebra. We prove that for a large class of groupoids $G$ with abelian isotropy---including all Deaconu--Renault groupoids associated to discrete abelian groups---$M$ is a maximal abelian subalgebra. In the specific case of $k$-graph groupoids, we deduce that $M$ is always maximal abelian, but show by example that it is not always Cartan.

Characterized Subgroups of Topological Abelian Groups

A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (vn) of characters of X if H = {x ∈ X : vn(x) → 0 in T}. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class of auto-characterized groups (namely, the groups that are characterized subgroups of themselves by means of a sequence of non-null characters); in the case of locally compact abelian groups, these are proven to be exactly the non-compact ones. As a by-product of our results, we find a complete description of the characterized subgroups of discrete abelian groups.

Locally Quasi-Convex Compatible Topologies on a Topological Group

For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G,\(\widehat{G})\) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (H) of (G) that are easier to handle and show that \(\mathscr{C} (H)\) and \(\mathscr{C} (G/H)\) are large and embed, as a poset, in \(\mathscr{C}(G,τ)\). Important special results are: (i) if \(K\) is a compact subgroup of a locally quasi-convex group \(G\), then \(\mathscr{C}(G)\) and \(\mathscr{C}(G/K)\) are quasi-isomorphic (3.15); (ii) if (D) is a discrete abelian group of infinite rank, then \(\mathscr{C}(D)\) is quasi-isomorphic to the poset \(\mathfrak{F}_D\) of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group \(G \) with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset \( \mathscr{C} (G) \) is as big as the underlying topological structure of (G,τ) (and set theory) allows. For a metrizable connected compact group \(X\), the group of null sequences \(G=c_0(X)\) with the topology of uniform convergence is studied. We prove that \(\mathscr{C}(G)\) is quasi-isomorphic to \(\mathscr{P}(\mathbb{R})\) (6.9).

Annihilator methods for spectral synthesis on locally compact Abelian groups

Spectral analysis and spectral synthesis study translation invariant linear function spaces on Abelian groups. Basic function classes for this study are the exponential monomials. These function classes have been investigated on discrete Abelian groups successfully using the annihilator method. In this paper we extend this technique to non-discrete locally compact Abelian groups.

Localization of Frames

Motivated by Balan, Casazza, Heil, and Landau's results on localization of frames in a separable Hilbert space ℋ, we investigate the transitivity of localization relation of the frame ℱ = {f i } i∈I and sequence ℰ = {e j } j∈G in ℋ with respect to the associated map a: I → G, where G is a discrete abelian group and I is a index set. We also study some properties of ℓ p -column decay, ℓ p -row decay, weak homogeneous approximation property, and strong homogeneous approximation property of frame ℱ = {f i } i∈I , and sequence ℰ = {e j } j∈G , with respect to the associated map a.

On the stability of sparse convolutions

We give a stability result for sparse convolutions on ℓ 2 ( G ) × ℓ 1 ( G ) for torsion-free discrete Abelian groups G such as Z . It turns out, that the torsion-free property prevents full cancellation in the convolution of sparse sequences and hence allows to establish stability, that is, injectivity with an universal lower norm bound, which only depends on the support cardinalities of the sequences. This can be seen as a reverse statement of the Young inequality for sparse convolutions. Our result hinges on a compression argument in additive set theory.