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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.

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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 ...
Isochron's user avatar
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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 ...
Focaccia's user avatar
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1 answer
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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 ...
Focaccia's user avatar
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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 $\widehat{\langle U\rangle}$ ...
Pedro Lourenço's user avatar
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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 ...
Pedro Lourenço's user avatar
3 votes
0 answers
55 views

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 ...
Pedro Lourenço's user avatar
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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 ...
Rolan Rial's user avatar
5 votes
2 answers
124 views

Normal character on a group von Neumann algebra

For a locally compact group $G$, I will denote by $L(G)$ its group von Neumann algebra, which is the von Neumann algebra acting on $L^2(G)$, generated by the image of left regular representation $\...
Mogget's user avatar
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Existence of a real continuous function with compact support and Haar integral $1$ over a topological group

I was studying Chapter XII of Lang's book Real and Functional Analysis and came across the following observation (page 315): Let $h\in C_c(G)$ be a positive function such that \begin{align*} \int_Gh\,...
Matheus Frota's user avatar
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Is the strict topology stronger than the weak* topology on the Fourier-Stieltjes algebra?

Let $G$ be a locally compact group and $B(G)$ its Fourier-Stieltjes algebra. It can be defined as either the dual of the group C$^*$-algebra $C^*(G)$ or the linear span of continuous positive definite ...
user680089's user avatar
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Can $C_C(X \times G)$ be identified as a subspace of $C_C(G, C_0(X))$?

I had previously asked Given locally compact spaces X and Y, must the following hold $C_b(X \times Y) \subseteq C_b(X, C_b(Y))$? We may assume $C_b(Y)$ is endowed with the topology of the sup-norm. ...
Brian's user avatar
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4 votes
2 answers
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Haar Measure of Product of Compact Sets

This is from Bachman's Harmonic Analysis book, exercise 8.3 Let $\mu$ be a Haar measure in $G$ and let $E_1$ and $E_2$ be two compact subsets of $G$ such that $\mu(E_1)=\mu(E_2)=0$. Does this imply ...
Vinay Deshpande's user avatar
3 votes
1 answer
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locally profinite and profinite group

Let $G$ be a locally profinite group (so a topological group such that every open neighbourhood of the identity in $G$ contains a compact open subgroup of $G$). It is well know that $$G \text{ is ...
Mario's user avatar
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Covering balls by balls of half radius in compact metric space (or, comparing Haar measure between balls of comparable radius)

Let $(X,d)$ be a compact metric space. Given $r>0$ I can obviously cover any ball $B(x,r)$ by finitely many balls $B(x_j,r/2)$, $j=1,...,k(r)$, by compactness. My question is whether there is ...
User's user avatar
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Integration over the general linear group over $\mathbb Q_p$

Before asking, I apologize if this question is duplicated. It seems a sort of basic stuff and I tried to find the reference, but I couldn't find it(only the statement without the proof appears in the ...
LWW's user avatar
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Identify Closed Subgroup in Compact Abelian Group

I came across this problem that given a subgroup $L$ of a compact abelian group $A$, how can we tell if it is closed? I think Pontryagin duality might be helpful. Identify $A$ with its double dual $\...
JNF's user avatar
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0 answers
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Tannaka–Krein duality in non-compact case

The Tannaka–Krein duality provides a way to reconstruct a group (up to isomorphisms) from the category of linear representations of that group. In physics, this duality is sometimes used as a ...
Weier's user avatar
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Two notions of $G$-integrable topological vector spaces

I've been working through Hochschild and Mostow's paper and they have the following definition of a $G$-integrable topological vector space $A$ where $G$ is a locally compact topological group. Below, ...
Steve's user avatar
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Triple Product Formula on $K = SU(2)$

Let $K = SU(2) = \{ k[\alpha ,\beta] | \alpha ,\beta \in \mathbb{C}, |\alpha |^2 + |\beta |^2 = 1 \} $ with $$ k [ \alpha , \beta ] = \begin{pmatrix} \alpha & \beta \\ - \overline{\beta} & \...
Misaka 16559's user avatar
2 votes
1 answer
243 views

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 ...
Mogget's user avatar
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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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1 answer
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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
0 answers
35 views

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
3 votes
1 answer
75 views

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 $...
user avatar
1 vote
1 answer
63 views

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
152 views

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
1 vote
1 answer
78 views

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
330 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
6 votes
1 answer
269 views

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
1 vote
0 answers
53 views

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
2 votes
0 answers
71 views

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
1 vote
1 answer
82 views

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
1 vote
1 answer
77 views

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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5 votes
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
9 votes
0 answers
322 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
0 answers
120 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
297 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
285 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
0 votes
1 answer
133 views

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
1 vote
0 answers
178 views

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
0 votes
0 answers
47 views

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
1 vote
0 answers
71 views

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
2 votes
0 answers
77 views

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 ...
Suzet's user avatar
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0 votes
1 answer
116 views

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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0 answers
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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
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3 votes
1 answer
180 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
2 votes
1 answer
50 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
173 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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1 answer
52 views

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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2 votes
0 answers
122 views

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 ...
kringelton4000's user avatar

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