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Questions tagged [locally-compact-groups]

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 relevant other tags whenever appropriate in order to reflect the main intentions of the question in the tags.

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 relevant other tags whenever appropriate in order to reflect the main intentions of the question in the tags.

436 questions
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Translation-invariant metric on locally compact group

Let $G$ be a locally compact group on which there exists a Haar measure, etc.. Now I am supposed to take such a metrisable $G$, and given the existence of some metric on $G$, prove that there exists a translation-invariant metric, i.e., a metric $d$…
Lit
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4
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1 answer

Why is $L^{1}(G)$ unital if and only if $G$ is discrete?

I've seen it stated in several sources and lecture notes for Abstract Harmonic Analysis that for a locally compact group $G$, $L^{1}(G)$ is unital if and only if $G$ is discrete. What about the locally compact group $\mathbb{T} =…
roo
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2
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0 answers

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 explain in detail? I can't seem to find elementary…
Jack Yoon
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2
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1 answer

Characterisation of subgroups that give compact quotients

Let $G$ be a locally compact Hausdorff group and $H \le G$ a closed subgroup. Are there properties of $H$ that implies that the quotient $G/H$ is compact? My guess would be that $H$ needs to be sufficiently spread out, but I don't have a nice way of…
user920957
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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 group. Any help is appreciated. Thank you.
user100478
  • 179
1
vote
0 answers

Query about the justification of a statement in Ihara's paper "On discrete subgroups of the two by two projective linear group over p-adic fields"

In the paper "On discrete subgroups of the two by two projective linear group over p-adic fields" https://projecteuclid.org/download/pdf_1/euclid.jmsj/1260541107 in section 1 when presenting the axiom (G, l, I), he says that the axiom he gives is…
Rupert
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1
vote
1 answer

Disjoint left translates of a function from a non-compact locally compact group to R with compact support.

I'm having trouble trying to prove an unjustified (and probably obvious!) statement in an academic paper. $G$ is a (locally compact) topological group which is not compact. $f$ is a continuous function $G \rightarrow \mathbb{R} $ with compact…
JJ Harrison
  • 450
0
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1 answer

Module function on automorphisms of discrete locally compact group

Let $G$ be a discrete locally compact group and let $\alpha: G \to G$ be an automorphism. Show that the module $\rm{mod}_{G}(\alpha)$ is 1. In the case of a locally compact field $k$ and $\alpha$ being multiplication by an element $a$ with…
Jacob Bond
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0
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Containing a net with a compact set.

Let $G$ be a locally compact group and let $(x_{\alpha})$ be a convergent net, say to $x$, in $G$. Is it possible to construct a compact subset $K$ of $G$ which contains each $x_{\alpha}$ and $x$?
roo
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0
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Convention for locally compact groups?

$\bf{\text{Suppose I find the phrase:}}$ Let $G$ is a locally compact group, and $\mathcal{U}$ a basis of neighborhoods of $1$. $\bf{\text{Question:}}$ Is it a convention to automatically take each $U\in\mathcal{U}$ to be compact? Clearly this can…
roo
  • 5,598
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Example of a locally compact but not locally connected group

Does someone know any example of a locally compact but not locally connected group? Thank you
user833135
0
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1 answer

Subset sums in an LCA group

Let $G$ be LCA, compact, $F\subseteq G$, $F+F=G$, $F$ open. Is it true that for each $g\in G$, $g+F\cap F\neq \emptyset$? In particular, this is equivalent to $F-F=G$. I've tried a bunch of examples for $S^1$. There seems to be a connection to the…
Ashwin Trisal
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topological vector space over locally compact field

Let $K$ be a nondiscrete locally compact topological field, and $V$ be a finite dimensional topological vector space over $K$ and $\{ v_1, \cdots ,v_n\}$ be a base of $V$. $\varphi : K^n \rightarrow V$ is a linear map $$ \varphi(x_1, \cdots , x_n)…
Kitamado
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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 underlying additive group of $K$. The theorem (see…
Olórin
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