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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Type I group von Neumann algebra

Let $G$ be a locally compact group. One says that $G$ is type I if its full group $C^*$ algebra is type I. Since being type I passes to quotients and generated von Neumann algebras, if $G$ is type I ...
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Quotient Integral formula: functions of compact support

Question: Let $G$ be a locally compact group and $H$ be a closed subgroup of $G$. given $0 \neq g \in C_c(G/H)$, the support of $g$, $C:= \text{spt}(g) \subset G/H$ is compact. Then, there exists ...
L-JS's user avatar
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Why do characters fail to characterize non-abelian LCH-groups?

For any locally compact Hausdorff abelian group (LCA group) $A$, a character $\xi\colon A\mapsto\mathbb{T} $ is definined as a continuous group homomorphism to the unit circle $\mathbb{T}\subseteq\...
Sidney Neffe's user avatar
2 votes
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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) ...
Pedro Lourenço's user avatar
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Pontrjagin duality for a topological ring

Let $R$ be a locally compact topological ring, and let $S$ be its Pontrjagin dual under addition. For instance, ℤ/nℤ is Pontrjagin dual to the ring ℤ/nℤ $\mathbb{Q}$/ℤ is Pontrjagin dual to the ring $...
Kind Bubble's user avatar
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1 answer
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Non-discrete LCAG's whose closed non-trivial subgroups are open

Let G be a locally compact abelian group and suppose that for any non-trivial closed subgroup H of G, H is open. It also makes sense to suppose that G is non-discrete (since in that case this ...
Pedro Lourenço's user avatar
3 votes
1 answer
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When is the intersection of all open normal subgroups equal to the connected component at the identity?

I've managed to show that for locally compact abelian groups, the connected component at the identity $G_0$ is equal to the intersection of all open subgroups of G, since we have that $$A(G_0) = \...
Pedro Lourenço's user avatar
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Subgroups of self dual groups

Let G be an infinite locally compact abelian group that is isomorphic to its own dual. If H is a closed subgroup of G, is it necessarily true that $H \cong \widehat{G/H}$? I ask because in the case of ...
Pedro Lourenço's user avatar
9 votes
1 answer
252 views

Weird duality between integers and p-adic integers

The integers are, up to isomorphism, the unique infinite discrete abelian group that is isomorphic to all its non-trivial subgroups. This can be seen easily by Pontryagin Duality: It's equivalent to ...
Pedro Lourenço's user avatar
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Is Fourier transform on compact group dependent on representatives of unitary equivalence class?

I'm trying to understand the Fourier transform on compact groups but run into technical problems regarding the equivalence classes of unitary representations. Here is my train of thought: $G$ is a ...
mateusz's user avatar
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Real Elliptic curves as compact abelian groups

It's well known that an elliptic curve of the form $y^3 = x^3 +ax +b$ admits a group structure, as long as a point at infinity $O$ is added to serve as identity. If we look at a real elliptic curve as ...
Pedro Lourenço's user avatar
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Something similar to the Minkowski's lattice theorem for the locally compact groups (problem with an exercise)

There is the following problem in the book "Fourier Analysis on Number Fields" by D. Ramakrishnan and R. Valenza: Let $G$ be a locally compact abelian subgroup with Haar measure $\mu$, and $\...
filipux's user avatar
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Is norm-continuous representation factored through a Lie quotient group?

Let $G$ be a locally compact group, and $X$ a Hilbert space. A unitary representation $\varphi:G\to B(X)$ is said to be norm-continuous, if it is continuous with respect to the norm in $B(X)$: $$ x_i\...
Sergei Akbarov's user avatar
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1 answer
101 views

Product of locally Borel subsets is locally Borel

Let $X$ be a locally compact Hausdorff space with a fixed Radon measure $\mu$. A subset $E\subseteq X$ is called locally Borel iff $E \cap A$ is a Borel subset of $X$ for every Borel subset $A$ of ...
Andromeda's user avatar
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Construct an invariant mean from an invariant functional

I am reading a paper, and I got stuck at an implication, which the authors believed to be natural, and hence did not provide a reasoning. Unfortunately, I can not see immediately why that implication ...
Carl Butcher's user avatar
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1 answer
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Is the Pontryagin map uniformly continuous?

Let $A$ be a locally compact abelian (LCA) group. We shall denote its dual group by $\widehat{A}$. Pontryagin duality states there is a canonical isomorphism $\delta: A \to \widehat{\widehat{A}}$ of ...
stoic-santiago's user avatar
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Is integration with respect to a Radon measure a normal weight on $L^\infty(X)$?

This might be trivial to experts in von Neumann algebra theory, but here goes. Let $\mu$ be a Radon measure on a locally compact Hausdorff space. Consider the weight $$\varphi: L^\infty(X, \mu) \to [0,...
Andromeda's user avatar
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1 answer
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Are Haar measures localizable?

I'm trying to prove that Haar measures are localizable. we know that Haar measures are decomposable ( see Haar measures are decomposable) in the sense that: A measure space $(X,\mathfrak{M},\mu)$ is ...
Amirhossein Haddadian's user avatar
8 votes
0 answers
239 views

Haar measures are decomposable

In the real analysis book by Folland, section $11.1$ exercise $9$ have been come that: if $G$ is a locally compact topological group with Haar measure $\mu$, then $\mu$ is decomposable. A measure ...
Amirhossein Haddadian's user avatar
2 votes
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104 views

If the topology of locally compact topological group G is not discrete, then haar measure of singlton is zero $\mu(\{x\})=0$

in folland-real analysis,chapter 11.1, exercise $9$ have been come that: if the topology of locally compact topological group G is not discrete, then haar measure of singlton is zero meaning that $\...
Amirhossein Haddadian's user avatar
1 vote
1 answer
187 views

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 $B\subseteq E$...
Amirhossein Haddadian's user avatar
4 votes
3 answers
225 views

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 ...
stoic-santiago's user avatar
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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 $\mathbb{R}^n$ ...
chaki chaki's user avatar
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Haar measures are saturated

We know that a saturated measure is a measure space $(X,\mathcal{M},\mu)$ such that $\mu:\mathcal{M}\rightarrow [0,\infty]\;$ and $\;\mathcal{M}=\widetilde{\mathcal{M}}\;$ where $\;\widetilde{\...
Amirhossein Haddadian's user avatar
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Abstract representation of a locally compact topological group which is not topological

Let $G$ be a locally compact topological group, and $V$ be a locally convex topological vector space over $\mathbb{C}$. An abstract representation of $G$ is a homomorphism $\rho\colon G\rightarrow \...
Gus Schmidt's user avatar
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Pure open subgroups of locally compact abelian groups

A subgroup $H$ of $G$ is called pure subgroup if $nH=H\cap nG$ for all $n\in\mathbb{Z}$. Let $H$ be a closed subgroup of nondiscrete locally compact abelian group $G$. An easy calculation shows that ...
Aliakbar's user avatar
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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) ...
stoic-santiago's user avatar
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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 ...
Unknown x's user avatar
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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 ...
OrthoPole's user avatar
3 votes
1 answer
134 views

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. ...
stoic-santiago's user avatar
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A question about pro-$\mathcal X$-group

This question concerns the following lemma of this paper: Lemma 2. Let $\mathcal X_1$,..., $\mathcal X_n$ be classes of finite groups closed with respect to normal subgroups and subdirect products ...
Meisam Soleimani Malekan's user avatar
2 votes
1 answer
36 views

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 ...
user920957's user avatar
1 vote
1 answer
78 views

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 ...
jdxoxo's user avatar
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Group action on a topological space and Haar integral (A question on a proof in a Palais' paper)

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 ...
Allotrios's user avatar
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Is assigning the central character a quotient map?

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 ...
Zeonive's user avatar
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3 votes
1 answer
131 views

Topological semi-direct products

In Kaniuth, Taylor, Induced representations of locally compact groups on pages 9-10 it's claimed that if $G$ is a locally compact group with closed subgroups $N,H$, with $N$ normal in $G$, with $N\cap ...
Matthew Daws's user avatar
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1 vote
1 answer
95 views

Question concerning positive Weyl chamber

I would like to ask for a hint for exercise 22.5 in Bump's book "Lie groups". The setting is as follows: Let $G$ be a (semisimple, connected, simply connected) compact Lie group, choose a ...
Mogget's user avatar
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5 votes
1 answer
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Considering Haar integral as an element of an algebra

For a Hilbert space $H$, a locally compact (Hausdorff) group $G$ and its (left) Haar measure $\mu$, it follows by the Riesz representation theorem that we may identify an operator $\int_G f(s)d\mu(s) \...
Deracless's user avatar
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$G$ be loacally compact abelian group with $G,\hat{G}$ both sigma compact. Let $f\in L^2(G)$ and $\psi_n(x)=\int_{C_n}\hat{f}(\chi)\chi(x)\ d\chi$.

Let $G$ be a locally compact abelian, $T_2$ group such that $G,\hat{G}$ both are $\sigma$-compact with $G=\bigcup K_n$ and $\hat{G}=\bigcup C_n$ where $K_n,C_n$ are increasing compact subsets of $G$ ...
MathBS's user avatar
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Is the Haar measure of the boundary of an open subset of the $k$-torus zero?

I have just recently started reading the basics of topological groups and Haar measures, and have become really curious about when the boundary of a non-empty open subset $U$ of a compact abelian ...
asrxiiviii's user avatar
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1 answer
37 views

Existence of a certain type of function on locally compact groups

I have seen it stated that on a locally compact group $G$ with $\mu$ its (left) Haar measure, there exists a positive, compactly supported function $f\in C_c(G)$ with $\int_G f(s)d\mu(s)=1$ satisfying ...
Deracless's user avatar
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2 votes
1 answer
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When does Inverse Fourier transform look close to a positive definite function?

Let $G$ be a commutative locally compact group, and $\hat{G}$ be its dual group, consisting of all continuous characters (continuous homomorphisms from $G$ to the circle group $\mathbb{T}$) . I can ...
Carl Butcher's user avatar
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71 views

Exercise 10 in section 11, chapter V, Conway's functional analysis.

On page 163, section 11, chapter V, Conway's functional analysis. Let $G$ be a locally compact group and $f\in\mathrm{C}_b(G)$. We say $f$ is almost periodic if \begin{equation} \mathcal{O}_f\,\colon\...
Sunny. Y's user avatar
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Let $G$ be locally compact abelian, $T_2$ and $f\in L^1(G)$. Define $\mu(A)=\int_A f(x)\ dx$. Prove $\lVert \mu\rVert=\lVert f\rVert_1$

Here $\mu$ becomes a complex measure and $\lVert \mu\rVert =|\mu|(G)$ is the total variation norm of $\mu$. We have to show $|\mu|(G)=\int\limits_G |f(x)|\ dx$ Let $\{A_n\}$ be a partition of $G$. ...
DeltaEpsilon's user avatar
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Can a complex Radon measure be approximated by compactly supported Radon measures?

Let $G$ be an (abelian) locally compact Hausdorff group. Consider the following fragment from Folland's text "A course in abstract harmonic analysis" (second edition, p102). Why is the ...
Andromeda's user avatar
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2 votes
2 answers
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Why is $\phi_\mu(x)= \int_{\widehat{G}} \xi(x)d\mu(\xi)$ a continuous map?

Let $G$ be a locally compact Hausdorff abelian group. Let $\widehat{G}$ be its dual group, consisting of the unitary characters $G \to \mathbb{T}$. If $\mu \in M(\widehat{G})$ (= complex Radon ...
Andromeda's user avatar
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Extension of a continuous and equivariant map to compactification

Given a locally compact (topological) group $G$ we denote by $C_b^{lu}(G)\subseteq C_b(G)$ the bounded continuous functions on $G$ such that $f\in C_b^{lu}(G)$ whenever the map $G\to C_b(G)$ given by $...
Deracless's user avatar
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0 answers
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Extending representations from a closed Lie subgroup - solution verification

Let $G$ be a compact Lie group and $H$ a closed Lie subgroup. Let $i : H \hookrightarrow G$ denote the inclusion. Then given any finite-dimensional representation $\rho : H \rightarrow \mathrm{GL}(V)$ ...
Neenu's user avatar
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Injectivity of divisible locally compact abelian groups

Are divisible locally compact abelian groups injective as objects of the quasi-abelian category of locally compact abelian groups ? At the very least, if $D$ is a divisible locally compact abelian ...
Dat Minh Ha's user avatar

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