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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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 ...
3
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1answer
23 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 ...
2
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1answer
26 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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1answer
39 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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22 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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14 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) ...
2
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1answer
25 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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1answer
46 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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1answer
45 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
14 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 ...
2
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1answer
61 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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1answer
45 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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42 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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58 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 ...
0
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1answer
26 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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41 views

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

Suppose I am looking for the Haar measure on $\displaystyle \prod _{n \in \mathbb N} G_i$ and have measure $\mu_i$ on each $G_i$. Is there a way I can use these to define the correct measure on the ...
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32 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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36 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
43 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 ...
4
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1answer
41 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
95 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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42 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
91 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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1answer
32 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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14 views

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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94 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, ...
0
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1answer
93 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
24 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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73 views

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
34 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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59 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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55 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 ...
2
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1answer
178 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 ...
2
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0answers
49 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 ...
2
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83 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 ...
4
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166 views

Uniqueness of Haar Measures

Haar measure on a LCA (locally compact abelian) group $G$ is said to be unique ... up to a scaling factor. This is not as elegant as it might be expected because this requires a choice of a unit ...
3
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1answer
177 views

Non-discrete locally compact group

Why every non-discrete locally compact group contains a nontrivial convergent sequence?
4
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1answer
133 views

Why are locally compact groups Weil complete?

Why are locally compact groups Weil complete? Note: A topological group $G$ is Weil complete if every left Cauchy net in $G$ is convergent. Thank you, and sorry if I have bad writing.
4
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1answer
193 views

Why do characters on a subgroup extend to the whole group?

As background, I am trying to do exercise 3.10 in Deitmar's "Principles of Harmonic Analysis." I can do most of the problem but I'm stuck on the third part proving surjectivity. Given a locally ...
3
votes
2answers
183 views

Shifting a function is continuous

I'm slightly puzzled by the following: if $g(t)$ is a function in $L^q(X)$ then we can show that $g(t-x)$ is continuous function of $t$, i.e. for $\varepsilon > 0$ we can find $\delta$ such that ...
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1answer
39 views

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 ...
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1answer
46 views

reference in Montgomery/ Zippin

In a paper, the authors use the reference [M–Z, Theorem in 4.13] where [M-Z] denote the book D. Montgomery and L. Zippin, Topological Transformation Groups. Interscience Publishers, New York–London, ...
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1answer
195 views

subgroup of connected locally compact group

I need a reference or a short proof for the following property: A nontrivial connected locally compact group $G$ contains an infinite abelian subgroup.
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440 views

A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
3
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1answer
210 views

If both $H$ and $G/H$ are locally compact then $G$ is locally compact (topological Group)

How do I prove this statement? Let $G$ be a Topological group and let $H$ be a subgroup of $G$, if both $H$ and $G/H$ are locally compact then $G$ is locally compact. (we will endow the set $G/H$ ...
3
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1answer
89 views

Characters being everywhere dense in the character group

Let $k$ be the completion of an algebraic number field at a prime divisor $\mathfrak{p}$. We note that $k$ is locally compact. Let $k^{+}$ be the additive group of $k$ which is a locally compact ...
5
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1answer
333 views

Subgroups of a vector space

I would like to have an overview of how a subgroup of a vector space over $\mathbb R$ of dimension $n$ can look like. Is there a complete classification available? I know that there are for examples ...
2
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1answer
119 views

Which of the following topological groups are polish or locally compact?

I want to show that the next groups are polish topological groups, which criteria should I use here? And also which are locally compact (same question)? The groups are: The group of permutations ...