# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### $SU_2(\mathbb{C})$ 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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### 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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### 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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### Is the group algebra 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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### 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 there ...
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### 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 $G$....
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### 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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### 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 ...
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### 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 ...
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### Non-discrete locally compact group

Why every non-discrete locally compact group contains a nontrivial convergent sequence?
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### 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.