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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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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24 views

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 ...
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47 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 ...
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49 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 ...
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27 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 ...
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12 views

Divisible, locally compact abelian groups

Let $G$ be a divisible, locally compact abelian group such that $G$ is a direct sum of a torsion group and a torsion-free group. Let $H$ be an open subgroup of $G$. Can we deduce that $H$ is a direct ...
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13 views

Discontinuous characters on locally compact abelian groups

I'm doing some reading out of Rudin's Fourier Analysis on Groups and Folland's Abstract Harmonic Analysis for research. Particularly I am on characters and dual groups and I've noticed that neither ...
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16 views

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 $x$?
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8 views

Kernel of a representation and central projections

Let $G$ be a locally compact group, and $B(G)$ be the Fourier-Stieltjes algebra of $G$. I'm trying to see why the following is true: let $\pi$ be a representation of $G$, the the kernel of $\pi$ in ...
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22 views

Haar measure of point sets

Let $G$ be a locally compact group with Haar measure $\mu$ (left or right doesn't matter to me). I know that the Haar measure is positive on open sets. What can be said about the Haar measure on ...
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1answer
38 views

A sequence of neighbourhoods with decreasing Haar measure in a non-discrete group

Let $G$ be a non-discrete (LCH) group. How can one find a sequence $(V_n)$ of compact, symmetric neighbourhoods of the identity element $1$ such that $\mu(V_n)\to 0$, where $\mu$ denotes the Haar ...
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2answers
57 views

Totally disconnected orbit spaces

Let $X$ be a totally disconnected $G$-space, where $G$ is a locally compact Hausdorff group. Is the orbit space X/G also totally disconnected? The same question for locally compact, Hausdorff, ...
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26 views

Unimodular groups

Let F be a non-archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series Fq((T)) over a finite field Fq (where q is a power of p). Is GL(n,F) ...
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1answer
47 views

Is the ring of integers of a local field an open subgroup?

I apologize if my question is a bit naive, but I don't have much experience in number theory and sometimes get very confused. Suppose $K$ is a non-archimedean local field (essentially a completion of ...
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2answers
44 views

Why do we want *unitary* representations of locally compact groups into $B(H)$?

This is related to a previous question of mine but I have a more philosophical issue with the material. Everywhere I have looked for representations of locally compact groups into $B(H)$, everyone ...
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1answer
32 views

Why is the image of $k^+$ dense in the character group?

Let $k$ be a local field, and consider the map $\phi: k^+ \hookrightarrow \widehat {k^+}$ given by $\eta \mapsto \eta X(\cdot)=X(\eta \cdot)$ where $X$ is a non-trivial character. Tate argues in his ...
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45 views

What is the cokernel of $\Bbb Q^{\text{disc}} \hookrightarrow \Bbb R$?

These two should be the standard examples for why Locally compact abelian groups are not an abelian categoty. The cokernel of any of these maps should is not a LCAG. $$\Bbb Q^{\text{disc}} ...
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66 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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21 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) ...
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1answer
28 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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52 views

Continuity of Modular Function

If $G$ is a locally compact group, there's a left invariant Borel measure on $G$, called Haar measure, which is unique up to multiplication by scalar. Denote it by $\mu$. For any $g\in G$, ...
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64 views

The dual of L^1(G) for a locally compact group G

I might be missing something, but most literature on topological groups and harmonic analysis that I've encountered mention that $L^\infty(G)$ can be naturally identified with the dual of $L^1(G)$ by ...
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1answer
15 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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What is the relationship between the spaces $\mathscr K (G)$ and $L^2(G)$?

The context is that $G$ is a locally compact Hausdorff group, $\mathscr K (G)$ is the space of continuous compactly-supported functions $G \to \mathbb C$ equipped with the inner product $(f|g) = \int ...
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79 views

Haar measure, convolution and involutions

I have some problems to follow the proof of the anti commutativity property of the convolution and involution operations defined using a Haar measure as presented in Pedersen's book "analysis Now", ...
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54 views

When does a left Haar measure on a locally compact group restrict to a left Haar measure on a locally compact subgroup?

Given a locally compact group which is not $\sigma$-compact, there exists a $\sigma$-compact subgroup $H$ of $G$ which is open and closed. A Remark in Folland's, A Course in Abstract Harmonic ...
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45 views

Construct a topological manifold which its open cover is locally finite but not globally

The whole question is like this: 1-4. Let M be a topological manifold, and let U be an open cover of M . (a) Assuming that each set in U intersects only finitely many others, show that U is locally ...
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87 views

Is a regular Borel measure on a locally compact space necessarily $\sigma$-finite?

I am trying to compile a proof of the uniqueness of Haar measure. Usually this is done by multiple-integral mumbo-jumbo, abusing left and right invariance of two potential measures and invoking ...
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1answer
27 views

Is group of rigid body motion compact?

I believe that group of rigid body motion is not compact. I mean all transformations in $R^3$ that preserve distance. But I need to know how to proof it? From where I should start to prove it?
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1answer
85 views

Is there a formula for the Haar measure on a product of groups?

Let $ (G_{n})_{n \in \mathbb{N}} $ be a sequence of locally compact topological groups with a corresponding sequence $ (\mu_{n})_{n \in \mathbb{N}} $ of Haar measures. Is there a way to construct a ...
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37 views

Prove uniqueness of Haar measure without using the integral?

All the literature I have seen proves the Haar measure is unique by first defining the Haar integral and then using Fubini's Theorem etc. to show that any two Haar integrals are scalar multiples of ...
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56 views

Convolution of $L^1(G)$ functions with elements of $M(G)$.

Let $G$ be a non-discrete locally compact group with left Haar measure $\mu$. There is an isometric embedding of $L^{1}(G)\to M(G), f\mapsto fd\mu$. Since $G$ is not discrete, the point-mass measure ...
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1answer
52 views

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 ...
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1answer
42 views

Showing a measure on a locally compact group is left invariant

I am trying to verify that the measure $\frac{1}{|x|}dx$ is a Haar measure on $\mathbb{R}\backslash \{0\}$. For every open interval $(a_{n},b_{n})\subset \mathbb{R}$ not containing $0$, I have that ...
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1answer
104 views

is the vector space $\mathbb{R}^\mathbb{N}$ locally compact?

is the vector space $\mathbb{R}^\mathbb{N}$ locally compact? for example, let $x=(x_1,x_2,....)$ any point of $\mathbb{R}^\mathbb{N}$ and let $V=[x_1-\epsilon,x_1+\epsilon] \times ...
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46 views

Lévy–Khintchin triplet

in a Lévy process with characteristic triplet $(\gamma,\sigma^2,\mu)$ what is the interpretation of the constants $\gamma$ and $\sigma^2$. Is $\gamma$ the expected value? For example, let $\Gamma$ a ...
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1answer
99 views

Can a locally compact group with closed singleton be countable but not discrete?

Problem: Prove if a locally compact group $(G,*)$ contains a closed singleton then it must be either discrete or uncountable Proof Given: Assuming $G$ is countable we can write $G = \displaystyle ...
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33 views

Locally compact infinite dihedral group

Is there any topology other than discrete which can be given to an infinite dihedral group to make it locally compact topological group? If no, why is it so?
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Question in “Fell” missing hypothesis?

I'm reading and studying Fell and Doran's book: Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles Chapter 3 is devoted to Locally compact groups. Exercise 35 says, ...
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118 views

Haar measure on O(n) or U(n)

Every locally compact group has left-invariants haar measures. In particular, the compact groups O(n) and U(n) have them. I was wondering if there is a realization of such a measure on these groups, ...
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1answer
100 views

How can this linear map be injective?

Studying the Peter-Weyl theorem, I've come across the following linear maps: $\theta_E: E'\otimes E \rightarrow$ Hom$(E,E)$ Where $ E'\otimes E$ denotes the tensor product of a finite-dimensional ...
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1answer
29 views

$SU_2(\mathbb{C})$ and the characters

i can prove that the irreducible characters $\chi_n$ of $SU_2(\mathbb{C})$ are equal to: $$\chi_n(e^{i\phi})=\frac{\sin((n+1)\phi)}{\sin(\phi)}$$ If i want to give the dimension of the representation ...
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1answer
27 views

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 ...
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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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1answer
36 views

Characterisation of the spectrum of certain unitary representations on $L^2(G)$

I have been informed of the existence of theorems in harmonic analysis that will allow me to calculate the spectrum of given unitary operators $L^2(G)$ where $G$ is a locally compact group. So far I ...
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60 views

The group algebra is separable

Let $G$ be a locally compact Hausdorff group. Is the group algebra $L^1(G)$ separable? Would you please help me or introduce references that can help me. Thanks
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58 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 ...
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1answer
209 views

Are nilpotent Lie groups unimodular?

The continuous homomorphism $\Delta:G \rightarrow \mathbb{R}^{\times}_+$ is defined by \begin{equation*} \int_G f(xy)dx = \Delta(y)\int_Gf(x)dx \end{equation*} where $dx$ is a left Haar measure on ...
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0answers
52 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 ...
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95 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 ...