# 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.

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### Proving Mutual Absolute Continuity of two Haar Measures

The following is a theorem from Dana William's book on Crossed Products of $C^{*}$-algebras, and I don't see how he verifies part of this claim. Let $G$ be a locally compact group with left Haar ...
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### Proof of the existence of the haar measure for compact group

I'm following these notes. In the proof of the Theorem 4 (page 3) there is an inequality that i am not able to verify. Let $G$ a compact group and $V$ an open neighbourhood of the identity. $A$ is ...
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### Existence of finite blocking sets for compact group

I'm following these notes. Let $G$ a compact group and $V$ an open neighbourhood of the identity. $A$ is called a $V$-blocking set if $A \cap gVh \neq \emptyset$ for all $g,h \in G$. Does always exist ...
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### Pre-Sheaf on locally compact abelian group

Let $G$ be a locally compact abelian group. We can define a pre-sheaf $\mathcal{F}$ on G by $$\mathcal{F}(U) := \widehat{\langle U\rangle}$$ for any open set $U$, where $\widehat{\langle U\rangle}$ ...
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### Locally compact group whose unitary irreducible reps are one dimensional

It's known that if G is a finite group such that all its irreducible unitary representations are one-dimensional, then G is abelian. This uses the fact that the left regular representation decomposes ...
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### Localization of locally compact commutative ring

If $A$ is a locally compact (Hausdorff) commutative ring and $S$ is a multiplicatively closed subset of $A$, is there any natural topology we can put on $A S^{-1}$ such that it also becomes a locally ...
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### If $G$ is a locally compact topological group, $H\triangleleft G$ with $H$ and $G/H$ compactly generated, then so is $G$.

I've been studying the next theorem stated in "Grupos Topológicos" by Tkachenko (1997): Let $G$ a locally compact group and $H\triangleleft G$. If $G/H$ and $H$ are compactly generated, then ...
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### Category of LCAG's "measures" the difference between the integers and the reals: what does this mean?

The Wikipedia article on Locally compact abelian Groups (https://en.m.wikipedia.org/wiki/Locally_compact_abelian_group) has the following excerpt in the Categorical properties section: Clausen (2017) ...
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### Are Haar measures semifinite?

We know that semifinite measure space $(X,\mathcal{M},\mu)$ is a measure space that for every measurable set $E\in\mathcal{M}$ with measure $\mu(E)=\infty$, there exist a measurable set $B\subseteq E$...
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### If $G$ is a locally compact group and $L^1(G)$ is unital, then $G$ is discrete.

The book Principles of Harmonic Analysis by Anton Deitmar and Siegfried Echterhoff outlines the following proof to show that if $G$ is a locally compact group and $L^1(G)$ is unital, then $G$ is ...
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### Subspace of Euclidean space

Consider the euclidean space $\mathbb{R}^n$. Consider a closed compact subset $A\subset\mathbb{R}^n$. For example, take $A=[a,b]^n$, with $0<a<b<\infty$. It is well know that $\mathbb{R}^n$ ...
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### Why are (Hausdorff) locally compact groups interesting?

I am currently taking classes in Combinatorics and Abstract Harmonic Analysis, which deal with the Fourier Analysis of functions on (mostly Hausdorff) locally compact topological (mostly abelian) ...
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### Does the vanishing of $\mathrm{Ext}^i_G(\rho,\pi)$ for every irreducible $\pi$ implies the same vanishing for more general representations $\pi$?

Let $G$ be a $p$-adic group, i.e., the group of $E$-points of some connected reductive group over a $p$-adic local field $E$. Let $\mathrm{Irr}(G)$ denote the set of equivalence classes of smooth ...
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### Why does "Any bounded sequence $(x_n)⊂H$ contains a sub-sequence $(x_{n_k} )$ such that $(Cx_{n_k} )$ converges."?

I was reading "Hilbert Space Operators in Quantum Physics" by Jiˇrí Blank • Pavel Exner • Miloslav Havlícek ˇ A linear everywhere defined operator $C$ on $H$ is said to be compact if it ...
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### Why the addition group of a local field is isomorphic to its character group? [duplicate]

In Tate's thesis, he claims that for a local field $k$ and any non-trivial character $\chi$ of its addition group $k^+$, the correspondence $\eta\mapsto\chi(\eta\xi)$ is an isomorphism, both ...
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### If a monothetic group is locally compact, then it is either discrete or compact

A topological group $G$ is called monothetic if it is Hausdorff and has a dense cyclic subgroup. I wish to prove that If a monothetic group is locally compact, then it is either discrete or compact. ...
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### 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 ...
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### Help with direct integral decomposition for locally compact groups

This might be a bit of a stupid question but I'm working on a question of Kirillov's "Elements of the Theory of Representations", where he proves that you can decompose a unitary ...
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The paper is: R. Palais, "On the existence of slices for actions of non-compact Lie groups", Annals of Mathematics, vol. 73, 1961. In the proof of Proposision 1.2.6, p. 301, $X$ is a ...
This is a follow-up question to this post. Let $G$ be a locally compact Hausdorff group, let $H$ be a closed central (i.e. $H\subset Z(G)$) subgroup and let $\iota:H\to G$ denote the inclusion. Then ...