Locally compact groups are among the main objects of study of abstract harmonic analysis. They arise e.g. in number theory, abstract Fourier analysis, representation theory, Lie theory, operator theory, etc. Their main property is the existence and uniqueness of Haar measure. Please combine with ...

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55 views

question regarding closure of symmetric neighborhood of e

I was going through the open mapping theorem (for topological groups) when I stumbled upon a topological property that I couldn't prove to myself. If I have a topological group, $G$ which is $\sigma$-...
3
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1answer
78 views

What are the continuous automorphisms of $\Bbb T$?

I wanted to check my reasoning on this problem. From standard Pontrjagin duality arguments, it's not hard to see that the continuous homomorphisms of the torus (to itself) are nothing more than the ...
2
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1answer
32 views

Show that a representative function on a profinite group factor through a surjection

Let $G$ be a compact group. A representative function $f\in\mathcal{C}(G,\mathbb{K})$ is a function such that $\dim\left(\operatorname{span}\left(Gf\right)\right)< \infty$. Remark that the ...
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1answer
36 views

Integral of $|f|$ outside a compact set

Let $G$ be a locally compact group. Given $f\in L^1(G)$ and $\epsilon>0$, how to show that there is a compact set $K\subset G$ such that $\int_{G\setminus K}|f|<\epsilon$?
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1answer
19 views

Example of non-amenable group which is the inverse limit of amenable groups

1) Does there exist a non-amenable locally compact group $G$ which is the inverse limit $\varprojlim G_i$ of amenable groups $G_i$? 2) Does there exist a non-amenable locally compact group $G$ which ...
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1answer
52 views

Two questions about increasing unions of compact subsets of a locally compact Hausdorff group.

I have two questions to ask related to my research. Question 1. Let $ G $ be a locally compact Hausdorff group. Is it possible that $ G $ is the union of a chain of compact subsets (ordered by ...
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1answer
42 views

Sigma-compact Polish groups

I would like to see an example of a sigma-compact Polish group which is not locally compact. I know that e.g. $l^{\infty}$ is a topological group which is sigma-compact but not locally compact. But ...
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1answer
28 views

Automophism of G and Haar measure

Let $G$ be a locally compact group (written additively), $\lambda$ an automophism of $G$, and $\alpha$ a Haar measure in $G$. As the Haar measure is unique up to factor constant, $\lambda$ transform $\...
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1answer
8 views

If $f(gK) \subset U$ find a compact neighborhood of $g$, $\overline{Q}$, such that $f(\overline{Q} K) \subset U$

sorry for my bad english. I am in a proof and I get stuck in the following step. Let $f : X \to Y$ be a continuous function between two topological spaces, $G$ a locally compact topological group, ...
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1answer
39 views

closed subset of locally compact

A space $X$ is said locally compact if for any $x\in X$ and for any neighbourhood $U$ of $x$ there is a compact neighbourhood $V$ such that $V\subseteq U$. Does closed subset of locally compact is ...
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1answer
34 views

How can we topologize the character group $\Gamma$ of a locally compact abelian group $G$, such that $\Gamma$ becomes a LCA group?

How can we topologize the character group $\Gamma$ of a locally compact abelian group $G$, such that $\Gamma$ itself becomes a LCA group ? I would really really appreciate if I can get a step by step ...
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1answer
18 views

Show $T:G\rightarrow L^1(G)$ by $y\mapsto f(y\cdot)$ is continuous for fixed $f\in L^1(G)$

Let $G$ be a locally compact group, and show $T_f:G\rightarrow L^1(G)$ by $y\mapsto f(y\cdot)$ is continuous for fixed $f\in L^1(G)$
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818 views

Positive definite function zoo

A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a coefficient of a unitary representation of $G$. For a definition and discussion of ...
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102 views

Reduced $C^*$-algebra of a direct product of locally compact groups

Is it true that $$C^*_r(G_1\times G_2)=C^*_r(G_1)\otimes_{\min}C^*_r(G_2)$$ for locally compact groups $G_1$ and $G_2$? I have managed to prove that it holds for discrete groups (see below), but as ...
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46 views

Functorial construction of the convolution algebra of measures on a group

Let $G$ be a lcoally compact group and $C_c(G) = \lim_{K \subset G} C(K)$ its space of continuous functions with compact support endowed with the topology of the limit of banach spaces $C(K)$ with $K$ ...
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0answers
64 views

Haar measure on a profinite group is the inverse limit of the counting measure on its quotients?

I've heard this a few times now, though I've never seen a precise result. I guess the precise statement would be close to: Let $N_i$ be a basis normal subgroup neighborhoods of the identity in a ...
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0answers
246 views

Uniqueness of Haar Measures

Haar measure on a LCA (locally compact abelian) group $G$ is said to be unique ... up to a scaling factor. This is not as elegant as it might be expected because this requires a choice of a unit ...
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170 views

Steinhaus theorem in topological groups

Let $(G,\cdot)$ be a locally compact Abelian topological group with Haar measure. It is known that if $B$ is a measurable subset of $G$ of finite and positive Haar measure then $int(B \cdot B)\neq \...
3
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0answers
83 views

Building up the Fourier inversion theorem on locally compact abelian groups

I'm reading through Folland's Abstract Harmonic Analysis and I've come to a bit of a road block with some machinery developed for the Fourier inversion theorem. We know that the Fourier transform of a ...
3
votes
0answers
100 views

Mapping $G$ into its group algebra as left multiplication. Continuous?

I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove: Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
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0answers
26 views

Conditions under which a convolution transformation is injective in the 1-d Torus

Let $X=[0,1)$ the 1-d torus. Given a bounded positive function $w\colon X\to\mathbb{R}$ with unit integral (I mean $w\geq 0$, $w\in L^\infty(X)$ and $\int_X w\; dx=1$), define \begin{align*} T_{w} ...
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0answers
25 views

*-representations, unitary representations, and adjunctions

I've been reading Folland's Abstract Harmonic Analysis, and I am currently in the section on the correspondence between unitary representations of a locally compact group $G$ and $*$-representations ...
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0answers
48 views

Norm Inequality (Vinogradov Notation)

I'm going through a proof of differentiability of fourier series on the d-dimensional torus and while proving the following inequality: $$ \sum_{n\in\mathbb{Z}^d}|a_n(f)|\ll_d\sqrt{\|f\|_2^2+\sum_{...
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0answers
41 views

Compact groups are Moore groups

A locally compact group $G$ is said to be a Moore group if each irreducible continuous unitary representation of $G$ is finite dimensional. I'm trying to see why a compact group $G$ would be a Moore ...
2
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0answers
106 views

Integration with values in a $C^*$-algebra

My question is quite specific to locally compact groups but I'm sure it can be generalised to locally compact Hausdorff spaces with a Borel measure. Let $G$ be a locally compact group and fix a Haar ...
2
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0answers
55 views

Locally compact group

I want/need to learn about locally compact groups in rigorous way. Mainly to completely understand Tate's thesis/Weil's basic number theory. Could you recommend a text or lecture note which would ...
2
votes
0answers
75 views

Induced measure on dual group and kernel of fourier transform

Let $G$ be a locally compact Hausdorff abelian group (LCA). Then the Fourier transform gives a map from functions on G to functions on $\hat G$ $$f\mapsto \hat f(\xi) = \int_G f(x)\Psi(x,\xi) \mu_G(x)...
2
votes
0answers
59 views

Molecular vibrations and a generalisation of Wigner's rule for (non-finite) compact groups

years student of mathematics and write my script for my bachelor. The topic is "Representations of groups and applications in physics". I understand the representations very good but now i want to ...
2
votes
0answers
117 views

Distributions over locally compact Abelian groups: when can they be Fourier transformed?

Pontryagin duality shows us every locally compact Abelian group---such as $\mathbb{R}^n$, $\mathbb{Z}$, the circle $\mathbb{R}/\mathbb{Z}$ or any finite Abelian group---has a Fourier transform. In the ...
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0answers
16 views

Is $\pi^1:C_c(G)\rightarrow \operatorname{End}(H)$ a homomorphism of the convolution algebra when $G$ is not unimodular?

Let $G$ be a Hausdorff locally compact group and $H$ a Banach space. Let $\pi:G\rightarrow \operatorname{GL}(H)$ be a representation and define $$\pi^1(\phi)v = \int_G\phi(x)\pi(x)vdx$$ for $v\in H$ ...
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45 views

Compact abelian group and Continuous functions

If $G$ is a compact abelian group, $\widehat{G}$ is the dual group of $G$,i.e. all the continuous homomorphism from $G$ to $S^1$,$S^1=\{z\in \mathbb{C}\big | |z|=1\}$. Show that the linear span of $\...
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vote
0answers
53 views

Second orthogonality relation for compact groups

On a finite group $G$, we have the second orthogonality relation $$\sum_{\chi\text{ irred.}} \chi(g)\overline{\chi(h)}=\begin{cases}|C_G(g)|&g=h\\0&\text{otherwise.}\end{cases}$$ for $g,h\in G$...
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0answers
31 views

Almost periodic compactification with the Gelfand-Naimark theorem

Could anyone please help me with a bibliographic reference presenting the almost periodic compactification of a topological group with the aid of the Gelfand-Naimark theorem? Rudin in "Fourier ...
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0answers
43 views

Explicit constructions of Haar measures?

I know how to build the Haar measure somewhat explicitely on Lie groups (via differential forms) and profinite groups (by using the lemma that open subsets of a profinite group are unions of cosets of ...
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vote
0answers
65 views

Locally compact spaces that are not first-countable and continuity of functions on locally compact groups and continuity of group representation

If $X$ is a topological space that is first-countable, then a function $f: X \to Y$ into another topological space $Y$ is continuous if and only if $f$ is sequential continuous. Only the implication ...
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0answers
121 views

Compact operators on $L^2(G)$ as a reduced cross product of $C_0(G)$ and $G$.

If any of the terminology is unclear then please don't hesitate to point it out. My question is: is it true that when $G$ is a locally compact second countable group then: \begin{equation*} C_0(G) \...
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0answers
96 views

Is the Hilbert-Smith conjecture still unsolved?

Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group. Is this conjecture still unsolved? Is there ...
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0answers
13 views

Irreducible representations of locally compact semigroups

What class of semigroups have finite dimensional irreducible representations. For example, If we have a compact groups $G$ then every continuous irreducible representation of $G$ is finite ...
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16 views

connected or compact Dirichlet domain

Let $G$ be a second-countable locally compact group and $d$ be a proper (i.e. bounded closed sets are compact) left invariant metric on $G$. Let $\Gamma$ be a lattice subgroup of $G$. Consider the ...
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24 views

$G$-invariant functions on Manifolds

In a paper I saw the following statement: Let $M$ be a connected symplectic manifold and $G$ be a compact Liegroup acting symplectically and hamiltonian on $M$. Let $\Phi \colon M \to \mathfrak{g^*}$ ...
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0answers
20 views

One-parameter subgroup K in G with pi(K) non-trivial where pi:G->G/H and given G/H has no small subgroups

This is exercise 12.12.8 in Dieudonne's Treatise On Analysis, Volume 2 Let G be a locally compact metrizable group H a closed normal subgroup of G G/H is not discrete and has no small subgroups ...
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24 views

Integration on compact group

Let $K$ be a compact topological group, and let $(V,\pi)$ be a continuous representation of $K$ over the complex field $\mathbb{C}$. Denote by $\mathrm{d}$ the Haar measure on $K$. If $v\in V$ ...
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35 views

From Folland Abstract Harmonic Analysis - Why is $\int f(x \xi) d\xi = \int f(x \xi^{-1}) \Delta_H(\xi^{-1}) d \xi$?

From Folland Abstract Harmonic Analysis - Why is $\int f(x \xi) d\xi = \int f(x \xi^{-1}) \Delta_H(\xi^{-1}) d \xi$? This is on page 57. Here is the notation: $H$ is a closed subgroup of a locally ...
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0answers
94 views

sequential continuity and countinuity

When we have two topological spaces, $\left(X, \tau_X\right)$ and $(Y, \tau_Y)$ it is easy to check that for $f: X \rightarrow Y$ continuity implies sequential continuity. I'm wondering what do we ...
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63 views

On the structure of non discrete locally compact topological (non-necessarily commutative) complete fields.

There is totally classic result about the structure of non discrete locally compact topological (non-necessarily commutative) fields $K$, whose proof uses the existence of the Haar measure on the ...