Discrete Abelian Groups Research Articles

A note on continuous stable sampling

As a starting point we assume to have a continuous frame in a Hilbert space with respect to a measure space. This frame inherits a unitary structure from a unitary representation of a locally compact abelian group in the Hilbert space. In this setting we state a continuous sampling result for the range space of the associated analysis frame operator. The data sampling are functions also defined by using the underlying unitary structure. The result is illustrated by using continuous frames in Paley–Wiener and shift-invariant spaces generated by translates of fixed functions. A sampling strategy working only for discrete abelian groups is also discussed.

Algebraic entropy for amenable semigroup actions

We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups S on discrete abelian groups A by endomorphisms; they extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case S=N. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group.For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a Bridge Theorem connects the algebraic entropy with the topological entropy of the dual action by means of Pontryagin duality, so that we derive an Addition Theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups.

Ternary n-Weak Amenability of Certain Commutative Banach Algebras and $$\hbox {JBW}^*$$-Triples

We show that a commutative unital Banach *-algebra is ternary n-weakly amenable when it is n-weakly amenable. We apply this result for a wide variety of commutative n-weakly amenable algebras such as for a convolution group algebra on a discrete abelian group and for a commutative unital $$\hbox {C}^*$$ -algebra. We also show that every commutative $$\hbox {JBW}^*$$ -triple is ternary n-weakly amenable. These results present a somehow unified extension of the previous ternary weak amenability results in the category of triple systems and n-weak amenability results in the category of Banach algebras.

Convolution Systems on Discrete Abelian Groups as a Unifying Strategy in Sampling Theory

A regular sampling theory in a multiply generated unitary invariant subspace of a separable Hilbert space $${\mathcal {H}}$$ is proposed. This subspace is associated to a unitary representation of a countable discrete abelian group G on $${\mathcal {H}}$$. The samples are defined by means of a filtering process which generalizes the usual sampling settings. The multiply generated setting allows to consider some examples where the group G is non-abelian as, for instance, crystallographic groups. Finally, it is worth to mention that classical average or pointwise sampling in shift-invariant subspaces are particular examples included in the followed approach.

Discrete convolution operators and Riesz systems generated by actions of abelian groups

We study the bounded endomorphisms of \(\ell ^2(G)\times \dots \times \ell ^2(G)=\ell _{N}^2(G)\) that commute with translations, where G is a discrete abelian group. It is shown that they form a C*-algebra isomorphic to the C*-algebra of \(N\times N\) matrices with entries in \(L^\infty ({\widehat{G}})\), where \({\widehat{G}}\) is the dual space of G. Characterizations of when these endomorphisms are invertible, and expressions for their norms and for the norms of their inverses, are given. These results allow us to study Riesz systems that arise from the action of G on a finite set of elements of a Hilbert space.

ARENS PRODUCTS, ARENS REGULARITY AND RELATED PROBLEMS

We study the second dual algebra of a Banach algebra and related problems. We resolve some questions raised by Ülger, which are related to Arens products. We then discuss a question of Gulick on the radical of the second dual algebra of the group algebra of a discrete abelian group and give an application of Arens regularity to Fourier and Fourier–Stieltjes transforms.

Riesz bases associated with regular representations of semidirect product groups

This work is devoted to the study of Bessel and Riesz systems of the type $$\big \{L_{\gamma }{\mathsf {f}}\big \}_{\gamma \in \Gamma }$$ obtained from the action of the left regular representation $$L_{\gamma }$$ of a discrete non abelian group $$\Gamma $$ which is a semidirect product, on a function $${\mathsf {f}}\in \ell ^2(\Gamma )$$. The main features about these systems can be conveniently studied by means of a simple matrix-valued function $${\mathbf {F}}(\xi )$$. These systems allow to derive sampling results in principal $$\Gamma $$-invariant spaces, i.e., spaces obtained from the action of the group $$\Gamma $$ on a element of a Hilbert space. Since the systems $$\big \{L_{\gamma }{\mathsf {f}}\big \}_{\gamma \in \Gamma }$$ are closely related to convolution operators, a connection with $$C^*$$-algebras is also established.

Predecessors of topologies of LCA groups

We show in this paper that for a non-compact LCA group (G,τ), τ has exactly 22k(G) predecessors in G2(G). This answers a problem posed in the literature affirmatively. Denote by N the class of all LCA groups (G,τ) such that P2(τ) is precompact. It is shown that for an LCA group G, G∈N if and only if G≅Rn×H, where n is a non-negative integer and H is an LCA group with an open compact subgroup N such that H/N∈M. As an application of this result, we extend a well known result on discrete abelian groups to the case of LCA groups. We show that if (G,τ) is a non-compact LCA group and σ is a predecessor of τ in G2(G), then the connected component of (G,τ) coincides with the connected component of (G,σ). It is also shown that for a non-compact LCA group (G,τ), if σ is a predecessor of τ in G2(G), then the equality ib(G,τ)=ib(G,σ) holds. This partially answer a question posed in the literature.

Discrete gauge symmetries and the weak gravity conjecture

In theories with discrete Abelian gauge groups, requiring that black holes be able to lose their charge as they evaporate leads to an upper bound on the product of a charged particle’s mass and the cutoff scale above which the effective description of the theory breaks down. This suggests that a non-trivial version of the Weak Gravity Conjecture (WGC) may also apply to gauge symmetries that are discrete, despite there being no associated massless field, therefore pushing the conjecture beyond the slogan that ‘gravity is the weakest force’. Here, we take a step towards making this expectation more precise by studying ℤN and ℤ2N gauge symmetries realised via theories of spontaneous symmetry breaking. We show that applying the WGC to a dual description of an Abelian Higgs model leads to constraints that allow us to saturate but not violate existing bounds on discrete symmetries based on black hole arguments. In this setting, considering the effect of discrete hair on black holes naturally identifies the cutoff of the effective theory with the scale of spontaneous symmetry breaking, and provides a mechanism through which discrete hair can be lost without modifying the gravitational sector. We explore the possible implications of these arguments for understanding the smallness of the weak scale compared to MPl.

Understanding of two-zero textures of neutrino mass matrices in minimal extended seesaw mechanism and their symmetry realization

We study the texture zeros of [Formula: see text] neutrino mass matrices [Formula: see text] in the minimal extended type-I seesaw (MES) mechanism, incorporating one extra gauge singlet field “[Formula: see text]”. The [Formula: see text] MES model deals with [Formula: see text] [Formula: see text], [Formula: see text] [Formula: see text] and [Formula: see text] mass matrix [Formula: see text] which couples the right-handed neutrinos and the singlet field “[Formula: see text]”. We carry out the mapping of all possible zero textures of [Formula: see text], [Formula: see text] and [Formula: see text] with the restriction to phenomenologically predictive cases having total eight zeros of [Formula: see text] and [Formula: see text] studied in the literature. If [Formula: see text], the sterile neutrino mass, is subject to any limit, further block diagonalization of [Formula: see text] shall not be allowed to reduce it to a [Formula: see text] matrix. In [Formula: see text] [Formula: see text] scenario, the study of texture zero is totally different and interesting. With this motivation, we consider the [Formula: see text] scheme where the digits of the pair represent the number of zeros of [Formula: see text] and [Formula: see text], respectively, along with the one/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. As the [Formula: see text] MES matrix is a matrix of rank 3, so we consider only those textures with two zeros which are of rank 3 whereby the number of feasible zero textures reduces to 12, out of 15. On realizing these 12 textures under MES mechanism with [Formula: see text] picture, we arrive at certain correlations for each texture. We examine the viability of each texture by scanning their respective correlations under recent neutrino oscillation data. Also, we discuss the interplay of Dirac and Majorana CP phases in determining the viability of a texture. The allowed two-zero textures are finally realized using a discrete Abelian flavor symmetry group [Formula: see text] with the extension of Standard Model to include some scalar fields.

Sampling Associated with a Unitary Representation of a Semi-Direct Product of Groups: A Filter Bank Approach

An abstract sampling theory associated with a unitary representation of a countable discrete non abelian group G, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples, the use of a filter bank formalism and the corresponding frame analysis allow for fixing the mathematical problem to be solved: the search of appropriate dual frames for ℓ 2 ( G ) . An example involving crystallographic groups illustrates the obtained results by using either average or pointwise samples.

On a Characterization Theorem for Probability Distributions on Discrete Abelian Groups

Let $X$ be a countable discrete Abelian group containing no elements of order 2, $\alpha$ be an automorphism of $X$, and $\xi_1$ and $\xi_2$ be independent random variables with values in the group...

Complex intertwinings and quantification of discrete free motions

The traditional quantification of free motions on Euclidean spaces into the Laplacian is revisited as a complex intertwining obtained through Doob transforms with respect to complex eigenvectors. This approach can be applied to free motions on finitely generated discrete Abelian groups: ℤm, with m ∈ ℕ, finite tori and their products. It leads to a proposition of Markov quantification. It is a first attempt to give a probability-oriented interpretation of exp(ξL), when L is a (finite) Markov generator and ξ is a complex number of modulus 1.

Spectral synthesis for the space of tempered solutions of a convolution system on discrete Abelian groups

We consider a problem of spectral synthesis in the topological vector space $${\mathcal{M}(G)}$$ of tempered functions on a discrete Abelian group G. It is proved that the space of tempered solutions of a convolution system on discrete Abelian groups admits spectral synthesis, that is the space of tempered solutions of a convolution system coincides with the closed linear span in $${\mathcal{M}(G)}$$ of all exponential monomial solutions of this system.

Filter banks on discrete abelian groups

In this work, we provide polyphase, modulation, and frame theoretical analyses of a filter bank on a discrete abelian group. Thus, multi-dimensional or cyclic filter banks as well as filter banks for signals in [Formula: see text] or [Formula: see text] spaces are studied in a unified way. We obtain perfect reconstruction conditions and the corresponding frame bounds in this particular context.

Three-dimensional anomalous twisted gauge theories with global symmetry: Implications for quantum spin liquids

Topological spin liquids can be described by topological gauge theories with global symmetry. Due to the presence of both nontrivial bulk deconfined gauge fluxes and global symmetry, topological spin liquids are examples of the so-called "Symmetry Enriched Topological phases" (SET). In this paper, we find that, in some twisted versions of topological gauge theories (with discrete Abelian gauge group $G_g$), implementing a global symmetry (denoted by $G_s$) is anomalous although symmetry charge carried by topological point-like excitations is normally fractionalized and classified by the second cohomology group. To demonstrate the anomaly, we fully gauge the global symmetry, rendering a new gauge theory that is not gauge invariant. Therefore, the SET order of the ground state is anomalous, which cannot exist in 3D system alone. Such anomalous state construction generalizes the "2D surface topological order" to 3D. A concrete example with $G_g=\mathbb{Z}_2\times\mathbb{Z}_4$ and $G_s=\mathbb{Z}_2$ is calculated.

Mean Proximality, Mean Sensitivity and Mean Li–Yorke Chaos for Amenable Group Actions

We consider mean proximality and mean Li–Yorke chaos for [Formula: see text]-systems, where [Formula: see text] is a countable discrete infinite amenable group. We prove that if a countable discrete infinite abelian group action is mean sensitive and there is a mean proximal pair consisting of a transitive point and a periodic point, then it is mean Li–Yorke chaotic. Moreover, we give some characterizations of mean proximal systems for general countable discrete infinite amenable groups.

On Heyde’s Theorem for Probability Distributions on a Discrete Abelian Group

Let X be a countable discrete Abelian group containing no elements of order 2. Let α be an automorphism of X. Let ξ1 and ξ2 be independent random variables with values in the group X and distributions μ1 and μ2. The main result of the article is the following statement. The symmetry of the conditional distribution of the linear form L2 = ξ1 + αξ2 given L1 = ξ1 + ξ2 implies that μ j are shifts of the Haar distribution of a finite subgroup of X if and only if α satisfies the condition Ker(I + α)= {0}. Some generalisations of this theorem are also proved.

Random walks on discrete Abelian groups

In the present paper we find necessary and sufficient conditions for recurrence of random walks on arbitrary subgroups of the group of rational numbers $\mathbb{Q}$.

On a characterization theorem for probability distributions on discrete Abelian groups

Пусть $X$ - счетная дискретная абелева группа, не содержащая элементов порядка 2, $\alpha$ - автоморфизм $X$, а $\xi_1$ и $\xi_2$ - независимые случайные величины со значениями в $X$ и с распределениями $\mu_1$ и $\mu_2$. Основной результат работы состоит в следующем. Для того чтобы из симметрии условного распределения линейной формы $L_2=\xi_1+\alpha\xi_2$ при фиксированной $L_1=\xi_1+\xi_2$ вытекало, что $\mu_j$ - сдвиги распределения Хаара некоторой конечной подгруппы группы $X$, необходимо и достаточно, чтобы $\alpha$ удовлетворял условию $\operatorname{Ker}(I+\alpha)=\{0\}$. Эта теорема является аналогом для дискретных абелевых групп известной теоремы Хейде, в которой гауссовское распределение на вещественной прямой характеризуется симметрией условного распределения одной линейной формы от независимых случайных величин при фиксированной второй. Доказаны также некоторые обобщения этой теоремы.