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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17 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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1answer
20 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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0answers
41 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
74 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
37 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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0answers
37 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
votes
0answers
123 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
150 views
Non-discrete locally compact group
Why every non-discrete locally compact group contains a nontrivial convergent sequence?
4
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1answer
98 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
votes
1answer
120 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
147 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 ...
1
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1answer
32 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
39 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
143 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.
6
votes
1answer
286 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
votes
1answer
146 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
votes
1answer
83 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
votes
1answer
218 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 ...
5
votes
2answers
962 views
Local-Global Principle and the Cassels statement.
In a recent article I have read, i.e. " Lecture notes on elliptic curves ", Prof.Cassels remarks in page-110 that
There is not merely a local-global principle for curves of genus-$0$, but
...
3
votes
2answers
106 views
Extensions and Kazhdan's Property (T)
Is Kazhdan's property (T) stable under extensions? i.e. if $G$ is an extension of
a group with property (T) by a group with property (T), does it follow that $G$ has property (T)?
3
votes
1answer
102 views
$\sigma$- compact clopen subgroup.
I am given $G$ locally compact group, and I want to show that there exists a clopen subgroup $H$ of $G$ that is $\sigma$-compact.
So here's what I did so far:
for $e \in U$, where $U$ is a nbhd of ...
4
votes
2answers
144 views
More on the versions of the Peter-Weyl theorem
The following three statements appear analogous:
For a finite group $G$, the group algebra $\mathbb C[G]$ decomposes as $\bigoplus_{V {\rm\ irred}} V^* \otimes V$.
(Peter-Weyl) For a compact group ...
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votes
1answer
327 views
Properties of Haar measure
Let $G$ be a locally compact group (but not discrete) and let $m$ be its left Haar measure. Is it true that $\forall \epsilon$ $\exists$ $C$ such that $C$ is a compact neighborhood of the identity and ...
3
votes
1answer
220 views
Left regular representation of $L^1(G)$ for a locally compact group $G$
Let $G$ be a locally compact group (not discrete) and let $L$ be the left regular representation of $A = L^1(G)$ on itself i.e. $L: A \to \mathcal{B}(A)$ where $L(f): A \to A$, $L(f)(g) = f*g$. I want ...
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vote
1answer
345 views
Young's inequality for discrete convolution
Young's inequality for convolution of functions states that for $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ we have
$$\|f\star g\|_r\le\|f\|_p\|g\|_q$$
for $p$, $q$, $r$ satisfying
...
10
votes
1answer
242 views
Stone-Weierstrass implies Fourier expansion
To prove the existence of Fourier expansion, I have to solve the following exercise, which supposedly follows from the Stone-Weierstrass theorem:
Let $G$ be a compact abelian topological group ...
10
votes
2answers
425 views
Translation-invariant metric on locally compact group
Let $G$ be a locally compact group on which there exists a Haar measure, etc..
Now I am supposed to take such a metrisable $G$, and given the existence of some metric on $G$, prove that there exists ...
2
votes
1answer
128 views
Surjective endomorphism preserves Haar measure
How to prove the following statement:
Let $G$ be a compact topological group and let $m$ be the Haar measure on it. Let $\varphi$ be a continuous endomorphism of $G$ onto $G$, i.e., the map $\varphi$ ...
3
votes
1answer
124 views
Why are characters required to be continuous?
I learned from several places that in defining a character of a topological group $G$, we often require it to be continuous, i.e. $\omega:G\to \mathbb{C}^{\times}$ is a continuous group homomorphism. ...
1
vote
1answer
115 views
Definition of admissible representation of the Weil group
Let $G$ be a locally profinite group, and $(\pi, V)$ be a representation of $G$ over a complex vector space $V$. Then the representation is called smooth if, for every $v$ in $V$, there is a compact ...
2
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0answers
123 views
Steinhaus theorem in topological groups
Let $(G,\cdot)$ be a locally compact Abelian topological group with Haar measure. It is known that if $B$ is a measurable subset of $G$ of finite and positive Haar measure then $int(B \cdot B)\neq ...
6
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1answer
498 views
Theorem of Steinhaus
The Steinhaus theorem says that if a set $A \subset \mathbb R^n$ is of positive inner Lebesgue measure then $\operatorname{int}{(A+A)} \neq \emptyset$. Is it true that also ...
6
votes
0answers
283 views
Positive definite function zoo
A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a coefficient of a unitary representation of $G$.
For a definition and discussion of ...
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1answer
200 views
Transitive group actions and homogeneous spaces
Given a topological group $G$ and a space $X$ with a transitive $G$ action, let $G_x$ be the isotropy group of a point. In Folland "A course in harmonic analysis", there is a statement that $X$ is ...
4
votes
1answer
373 views
Reference request: Fourier and Fourier-Stieltjes algebras
I would like to learn the basic theory of Fourier algebras and Fourier-Stieltjes algebras. In particular, I want to know how these two objects are defined in the case of not necessarily abelian ...
2
votes
2answers
110 views
reduced norm is proper
If $D$ is a central division algebra of dimension $n^2$ over a field $k$ we can consider the reduced norm $\nu : D \to k$, which satisfies $\nu^n = N_{D/k}$. In particular we get a group homomorphism ...
1
vote
1answer
159 views
Representation which have no unique decomposition into irreducible
What kind of examples of groups and representations should I keep in mind, which do not uniquely decompose into irreducible representations? I am mostly interested in characteristic zero ...
1
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1answer
122 views
Schwartz kernel theorem for induced representation/ Schur algebra for locally compact groups
Given a finite group $G$ and subgroups $H$ and $K$, and representation
$$\sigma: H \rightarrow GL(V_\sigma), \qquad \pi: K \rightarrow GL(V_\pi).$$
Now consider the space of functions $f: G ...
4
votes
1answer
101 views
Is there any relation between a group being unimodular and having equivalent uniform structures?
Recall: A topological group is said to have equivalent uniform structures if its left and right uniform structures coincide. A locally compact group is said to be unimodular if left Haar measures and ...
2
votes
1answer
170 views
Compact group actions and automatic properness
I am currently re-reading a course on basic algebraic topology, and I am focussing
on the parts that I feel I had very little understanding of. There is one exercise
in the chapter devoted to groups ...
