Haar measure and continuous representations of locally compact abelian groups

Jean-Christophe Tomasi

Displaying similar documents to “Haar measure and continuous representations of locally compact abelian groups”

On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths

Weidong Gao, Yuanlin Li, Pingping Zhao, Jujuan Zhuang (2016)

Colloquium Mathematicae

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Let G be an additive finite abelian group. For every positive integer ℓ, let $d i s c_{ℓ} (G)$ be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine $d i s c_{ℓ} (G)$ for certain finite groups, including cyclic groups, the groups $G = C ₂ \oplus C_{2 m}$ and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum...

On a translation property of positive definite functions

Lars Omlor, Michael Leinert (2010)

Banach Center Publications

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If G is a locally compact group with a compact invariant neighbourhood of the identity e, the following property (*) holds: For every continuous positive definite function h≥ 0 with compact support there is a constant $C_{h} > 0$ such that $\int L_{x} h \cdot g \leq C_{h} \int h g$ for every continuous positive definite g≥0, where $L_{x}$ is left translation by x. In [L], property (*) was stated, but the above inequality was proved for special h only. That “for one h” implies “for all h” seemed obvious, but turned out not to be obvious at...

The density of representation degrees

Martin Liebeck, Dan Segal, Aner Shalev (2012)

Journal of the European Mathematical Society

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For a group $G$ and a positive real number $x$ , define $d_{G} (x)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$ . We study the asymptotics of $d_{G} (x)$ for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups $G$ in characteristic zero, showing that either there exists $α > 0$ such that $d_{G} (x) > x^{α}$ for all large $x$ , or $G$ is virtually abelian (in which case $d_{G} (x)$ is bounded). ...

Subsets of nonempty joint spectrum in topological algebras

Antoni Wawrzyńczyk (2018)

Mathematica Bohemica

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We give a necessary and a sufficient condition for a subset $S$ of a locally convex Waelbroeck algebra $𝒜$ to have a non-void left joint spectrum $σ_{l} (S) .$ In particular, for a Lie subalgebra $L \subset 𝒜$ we have $σ_{l} (L) \neq \emptyset$ if and only if $[L, L]$ generates in $𝒜$ a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.

Completely bounded lacunary sets for compact non-abelian groups

Kathryn Hare, Parasar Mohanty (2015)

Studia Mathematica

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In this paper, we introduce and study the notion of completely bounded $Λ_{p}$ sets ( $Λ_{p}^{c b}$ for short) for compact, non-abelian groups G. We characterize $Λ_{p}^{c b}$ sets in terms of completely bounded $L^{p} (G)$ multipliers. We prove that when G is an infinite product of special unitary groups of arbitrarily large dimension, there are sets consisting of representations of unbounded degree that are $Λ_{p}$ sets for all p < ∞, but are not $Λ_{p}^{c b}$ for any p ≥ 4. This is done by showing that the space of completely bounded $L^{p} (G)$ ...

Self-small products of abelian groups

Josef Dvořák, Jan Žemlička (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let $A$ and $B$ be two abelian groups. The group $A$ is called $B$ -small if the covariant functor $Hom (A, -)$ commutes with all direct sums $B^{(κ)}$ and $A$ is self-small provided it is $A$ -small. The paper characterizes self-small products applying developed closure properties of the classes of relatively small groups. As a consequence, self-small products of finitely generated abelian groups are described.

On Beurling measure algebras

Ross Stokke (2022)

Commentationes Mathematicae Universitatis Carolinae

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We show how the measure theory of regular compacted-Borel measures defined on the $δ$ -ring of compacted-Borel subsets of a weighted locally compact group $(G, ω)$ provides a compatible framework for defining the corresponding Beurling measure algebra $ℳ (G, ω)$ , thus filling a gap in the literature.

Number of solutions in a box of a linear equation in an Abelian group

Maciej Zakarczemny (2016)

Colloquium Mathematicae

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For every finite Abelian group Γ and for all $g, a ₁, . . ., a_{k} \in Γ$ , if there exists a solution of the equation $\sum_{i = 1}^{k} a_{i} x_{i} = g$ in non-negative integers $x_{i} \leq b_{i}$ , where $b_{i}$ are positive integers, then the number of such solutions is estimated from below in the best possible way.

Stacks of group representations

Paul Balmer (2015)

Journal of the European Mathematical Society

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We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$ , the derived and the stable categories of representations of a subgroup $H$ can be constructed out of the corresponding category for $G$ by a purely triangulated-categorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods...

Extensions of generic measure-preserving actions

Julien Melleray (2014)

Annales de l’institut Fourier

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We show that, whenever $Γ$ is a countable abelian group and $Δ$ is a finitely-generated subgroup of $Γ$ , a generic measure-preserving action of $Δ$ on a standard atomless probability space $(X, μ)$ extends to a free measure-preserving action of $Γ$ on $(X, μ)$ . This extends a result of Ageev, corresponding to the case when $Δ$ is infinite cyclic.

The theory of reproducing systems on locally compact abelian groups

Gitta Kutyniok, Demetrio Labate (2006)

Colloquium Mathematicae

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A reproducing system is a countable collection of functions $ϕ_{j} : j \in$ such that a general function f can be decomposed as $f = \sum_{j \in} c_{j} (f) ϕ_{j}$ , with some control on the analyzing coefficients $c_{j} (f)$ . Several such systems have been introduced very successfully in mathematics and its applications. We present a unified viewpoint in the study of reproducing systems on locally compact abelian groups G. This approach gives a novel characterization of the Parseval frame generators for a very general class of reproducing systems...

Invariant subspaces for operators in a general II1-factor

Uffe Haagerup, Hanne Schultz (2009)

Publications Mathématiques de l'IHÉS

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Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace $𝒦 = 𝒦_{T} (B)$ affiliated with ℳ, such that the Brown measure of ${T |}_{𝒦}$ is concentrated...

Entropy of Schur–Weyl measures

Sevak Mkrtchyan (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Relative dimensions of isotypic components of $N$ th order tensor representations of the symmetric group on $n$ letters give a Plancherel-type measure on the space of Young diagrams with $n$ cells and at most $N$ rows. It was conjectured by G. Olshanski that dimensions of isotypic components of tensor representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to this family of Plancherel-type measures in the limit when $\frac{N}{\sqrt{n}}$ converges to a constant....

Endomorphism kernel property for finite groups

Heghine Ghumashyan, Jaroslav Guričan (2022)

Mathematica Bohemica

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A group $G$ has the endomorphism kernel property (EKP) if every congruence relation $θ$ on $G$ is the kernel of an endomorphism on $G$ . In this note we show that all finite abelian groups have EKP and we show infinite series of finite non-abelian groups which have EKP.

The max norm in $R^{n}$ -isometries and measure

Regina Cohen, James W. Fickett (1982)

Colloquium Mathematicae

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Characteristic points, rectifiability and perimeter measure on stratified groups

Valentino Magnani (2006)

Journal of the European Mathematical Society

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We establish an explicit connection between the perimeter measure of an open set $E$ with $C^{1}$ boundary and the spherical Hausdorff measure $S^{Q - 1}$ restricted to $\partial E$ , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and $Q$ denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of $E$ is less than or equal to $S^{Q - 1} (\partial E)$ up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...

Isolated subgroups of finite abelian groups

Marius Tărnăuceanu (2022)

Czechoslovak Mathematical Journal

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We say that a subgroup $H$ is isolated in a group $G$ if for every $x \in G$ we have either $x \in H$ or $〈 x 〉 \cap H = 1$ . We describe the set of isolated subgroups of a finite abelian group. The technique used is based on an interesting connection between isolated subgroups and the function sum of element orders of a finite group.

On a generalization of Abelian sequential groups

Saak S. Gabriyelyan (2013)

Fundamenta Mathematicae

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Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group ${(G, τ)}^{\land}$ is a dense -closed subgroup of the compact group ${(G_{d})}^{\land}$ , where $G_{d}$ is the group G with...

On the convergence and character spectra of compact spaces

István Juhász, William A. R. Weiss (2010)

Fundamenta Mathematicae

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An infinite set A in a space X converges to a point p (denoted by A → p) if for every neighbourhood U of p we have |A∖U| < |A|. We call cS(p,X) = |A|: A ⊂ X and A → p the convergence spectrum of p in X and cS(X) = ⋃cS(x,X): x ∈ X the convergence spectrum of X. The character spectrum of a point p ∈ X is χS(p,X) = χ(p,Y): p is non-isolated in Y ⊂ X, and χS(X) = ⋃χS(x,X): x ∈ X is the character spectrum of X. If κ ∈ χS(p,X) for a compactum X then κ,cf(κ) ⊂ cS(p,X). A selection of our...

Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space

Piotr Niemiec (2012)

Studia Mathematica

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For a linear operator T in a Banach space let $σ_{p} (T)$ denote the point spectrum of T, let $σ_{p, n} (T)$ for finite n > 0 be the set of all $λ \in σ_{p} (T)$ such that dim ker(T - λ) = n and let $σ_{p, \infty} (T)$ be the set of all $λ \in σ_{p} (T)$ for which ker(T - λ) is infinite-dimensional. It is shown that $σ_{p} (T)$ is $ℱ_{σ}$ , $σ_{p, \infty} (T)$ is $ℱ_{σ δ}$ and for each finite n the set $σ_{p, n} (T)$ is the intersection of an $ℱ_{σ}$ set and a $_{δ}$ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more...

Every $2$ -group with all subgroups normal-by-finite is locally finite

Enrico Jabara (2018)

Czechoslovak Mathematical Journal

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A group $G$ has all of its subgroups normal-by-finite if $H / H_{G}$ is finite for all subgroups $H$ of $G$ . The Tarski-groups provide examples of $p$ -groups ( $p$ a “large” prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a $2$ -group with every subgroup normal-by-finite is locally finite. We also prove that if $| H / H_{G} | \leq 2$ for every subgroup $H$ of $G$ , then $G$ contains an Abelian subgroup of index at most $8$ .

On the Davenport constant and group algebras

Daniel Smertnig (2010)

Colloquium Mathematicae

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For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S = g ₁ \cdot . . . \cdot g_{l}$ over G such that $(X^{g ₁} - a ₁) \cdot . . . \cdot (X^{g_{l}} - a_{l}) \neq 0 \in K [G]$ for all $a ₁, . . ., a_{l} \in K^{\times}$ . If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for...

Relations between Shy Sets and Sets of $ν_{p}$ -Measure Zero in Solovay’s Model

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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An example of a non-zero non-atomic translation-invariant Borel measure $ν_{p}$ on the Banach space $ℓ_{p} (1 \leq p \leq \infty)$ is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition " $ν_{p}$ -almost every element of $ℓ_{p}$ has a property P" implies that “almost every” element of $ℓ_{p}$ (in the sense of [4]) has the property P. It is also shown that the converse is not valid.