Questions tagged [locally-compact-groups]

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 relevant other tags whenever appropriate in order to reflect the main intentions of the question in the tags.

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 relevant other tags whenever appropriate in order to reflect the main intentions of the question in the tags.

395 questions
19
votes
1 answer

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 the measure of $C$ is less than$\epsilon$ ? I…
19
votes
2 answers

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 $\mathcal{L}^1(G, \mu)$. Define the function from $G$ to $\mathcal{L}^1(G)$…
15
votes
0 answers

Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$.

Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly supported functions $C_c(G)$ are dense in $L^2(G)$. In the book "Operator algebras, theory of $C^{*}$-algebras and von…
14
votes
2 answers

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 a translation-invariant metric, i.e., a metric $d$…
Lit
  • 303
  • 1
  • 6
14
votes
1 answer

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 $\operatorname{int}{(tA+(1-t)A)} \neq \emptyset$ for $t \in(0,1)$? It is…
12
votes
2 answers

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 it has a quantitative formulation ( and also, …
12
votes
1 answer

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.
11
votes
1 answer

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 with Haar measure $m$. Let $\hat G$ be the dual. The…
10
votes
1 answer

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 case of $\mathbb{R}$, it follows from …
10
votes
2 answers

A Radon measure on $G$ being left-invariant on a dense subgroup $H \subset G$ is a Haar measure on $G$.

Let $G$ be a locally compact group, $H$ a dense subgroup, and $μ$ a Radon measure on $G$ such that $μ(hA) = μ(A)$ holds for every measurable set $A ⊂ G$ and every $h ∈ H$. Show that $μ$ is a (left-invariant) Haar measure. Any help with this is…
9
votes
1 answer

Is a compactly supported function on a locally compact Hausdorff space uniformly continuous?

Consider the following fragment from Folland's book "A course in abstract harmonic analysis": Let $G$ be a locally compact Hausdorff group. I want to prove that the representation $$\pi: G \to B(L^2(\mu))$$ defined by $(\pi(x)f)(s) = f(x^{-1}s)$ is…
8
votes
1 answer

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 $$\frac{1}{p}+\frac{1}{q}=\frac{1}{r}+1.$$ Does this…
8
votes
1 answer

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, or its integral operator. Of course, right…
Henrique Tyrrell
  • 1,529
  • 11
  • 25
8
votes
0 answers

Finite measure fundamental domain for a discrete group implies it's a lattice

Here $G$ is a locally compact second countable topological group with left haar measure $\mu$, and $\varGamma$ is a discrete subgroup with a borel subset $\varOmega \subseteq G$ s.t. $G=\biguplus_{\gamma\in\varGamma}\varOmega\gamma$. Claim: If $…
8
votes
1 answer

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 compact abelian group $G,$ a closed subgroup $H,$ and…
mck
  • 1,236
  • 6
  • 18
1
2 3
26 27