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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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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190 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 ...
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191 views

Non-discrete locally compact group

Why every non-discrete locally compact group contains a nontrivial convergent sequence?
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141 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.
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1answer
227 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 ...
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2answers
193 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
47 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
199 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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1answer
512 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 $\mathcal{L}^1(G, \mu)$. ...
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1answer
221 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 $G/H$ ...
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1answer
94 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 ...
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1answer
356 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 ...
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1answer
134 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 ...
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2answers
1k 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 ...
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129 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)?
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145 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 ...
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177 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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1answer
559 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
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1answer
266 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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1answer
662 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 ...
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1answer
345 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 ...
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2answers
545 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
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1answer
187 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
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1answer
138 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. ...
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1answer
168 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 ...
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145 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 ...
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1answer
832 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
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0answers
429 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
256 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
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1answer
569 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
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2answers
161 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 ...
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1answer
207 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 ...
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1answer
147 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
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1answer
112 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 ...
3
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1answer
253 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 ...