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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Compact abelian group and Continuous functions

If $G$ is a compact abelian group, $\widehat{G}$ is the dual group of $G$,i.e. all the continuous homomorphism from $G$ to $S^1$,$S^1=\{z\in \mathbb{C}\big | |z|=1\}$. Show that the linear span of ...
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12 views

Continuous functions on compact group and uniformity

If $G$ is a compact abelian group and $f\in C(G)$. Then $\forall \epsilon >0$,there exists an open neighbourhood $U$ of $0\in G$, such that $\forall g\in G , \forall u_1,u_2\in U$, we have ...
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15 views

Conditions under which a convolution transformation is injective in the 1-d Torus

Let $X=[0,1)$ the 1-d torus. Given a bounded positive function $w\colon X\to\mathbb{R}$ with unit integral (I mean $w\geq 0$, $w\in L^\infty(X)$ and $\int_X w\; dx=1$), define \begin{align*} T_{w} ...
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Does a lattice in $SL_n(\mathbb R)$ which is contained in $SL_n(\mathbb Z)$ have finite index in $SL_n(\mathbb Z)$?

A lattice $H$ in a locally compact group $G$ is a discrete subgroup such that the coset space $G/H$ admits a finite $G$-invariant measure. I have read several places that any lattice H in ...
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$G$-invariant functions on Manifolds

In a paper I saw the following statement: Let $M$ be a connected symplectic manifold and $G$ be a compact Liegroup acting symplectically and hamiltonian on $M$. Let $\Phi \colon M \to \mathfrak{g^*}$ ...
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One-parameter subgroup K in G with pi(K) non-trivial where pi:G->G/H and given G/H has no small subgroups

This is exercise 12.12.8 in Dieudonne's Treatise On Analysis, Volume 2 Let G be a locally compact metrizable group H a closed normal subgroup of G G/H is not discrete and has no small subgroups ...
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Integration on compact group

Let $K$ be a compact topological group, and let $(V,\pi)$ be a continuous representation of $K$ over the complex field $\mathbb{C}$. Denote by $\mathrm{d}$ the Haar measure on $K$. If $v\in V$ ...
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42 views

Haar measure of quotient group

Suppose $G$ is a (Hausdorff) compact group with normalised Haar measure $\mu$, and that $H\trianglelefteq G$ is a closed normal subgroup. Is it true that the pushforward of $\mu$ to $G/H$ is the ...
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For a Borel action of a locally compact second countable group G on a standard Borel space S, are the orbits always Borel?

In his book "Ergodic Theory and Semisimple Groups" Robert Zimmer opens Chapter 2 by discussing the situation of a locally compact second countable group $G$ with a Borel action on a standard Borel ...
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how many group structures make $S^1$ a topological group?

let $S^1$ be the subspace of $R^2$ given the usual topology. How many group structures make $S^1$ a topological group?
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69 views

Reference for Harmonic Analysis?

I'm looking primarily for references for Harmonic Analysis. I'm mostly considering Doran&Fell or Deitmar, but I have access to lectures using Stein as well. The important thing is covering ...
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19 views

*-representations, unitary representations, and adjunctions

I've been reading Folland's Abstract Harmonic Analysis, and I am currently in the section on the correspondence between unitary representations of a locally compact group $G$ and $*$-representations ...
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50 views

Second orthogonality relation for compact groups

On a finite group $G$, we have the second orthogonality relation $$\sum_{\chi\text{ irred.}} \chi(g)\overline{\chi(h)}=\begin{cases}|C_G(g)|&g=h\\0&\text{otherwise.}\end{cases}$$ for $g,h\in ...
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26 views

Almost periodic compactification with the Gelfand-Naimark theorem

Could anyone please help me with a bibliographic reference presenting the almost periodic compactification of a topological group with the aid of the Gelfand-Naimark theorem? Rudin in "Fourier ...
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1answer
117 views

Pontryagin dual of the unit circle

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why? ...
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33 views

Explicit constructions of Haar measures?

I know how to build the Haar measure somewhat explicitely on Lie groups (via differential forms) and profinite groups (by using the lemma that open subsets of a profinite group are unions of cosets of ...
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Haar measure on a profinite group is the inverse limit of the counting measure on its quotients?

I've heard this a few times now, though I've never seen a precise result. I guess the precise statement would be close to: Let $N_i$ be a basis normal subgroup neighborhoods of the identity in a ...
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367 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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113 views

Connected Lie group is second countable?

I know this is true from various sources, unfortunately none of them give the full proof. I already have a start: Let $G$ be connected Lie Group. Choose $K$ to be any compact neighbourhood of the ...
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22 views

Lipschitz maps on locally compact groups

Suppose $G$ is a locally compact second countable group. This means that there exists a proper (closed bounded sets are compact) left invariant ($d(gx,gy) = d(x,y) \ \forall g,x,y \in G$) metric on ...
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324 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, ...
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Locally compact spaces that are not first-countable and continuity of functions on locally compact groups and continuity of group representation

If $X$ is a topological space that is first-countable, then a function $f: X \to Y$ into another topological space $Y$ is continuous if and only if $f$ is sequential continuous. Only the implication ...
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39 views

support compact modulo subgroup

I am studying (co)-induced representations of topological groups and I came across the following situation: $G$ is a topological group, $H$ a closed subgroup and $f\colon G\to W$ a set-theoretic map, ...
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39 views

LCA - groups under continuous homomorphisms

can someone help me out with this question? LCA stands for Locally compact Hausdorff abelian group. The question is posted in the attached image Let $A$ and $B$ be LCA-groups and $H$ a (not ...
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115 views

Finding an open set for a topological group

Let $G$ be a locally compact topological group, $K$ a compact subgroup and $\Gamma$ a discrete subgroup. I try to find a neighbourhood $U$ of the identity such that $\Gamma \cap UK = \Gamma \cap K$. ...
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135 views

Steinhaus theorem for topological groups

$G$ is a locally compact Hausdorff topological group, $m$ is a (left) Haar measure on $X$, $A$ and $B$ are two finite positive measure in $G$, that is $m(A)>0$, $m(B)>0$. My question is: Can ...
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57 views

What prevents the restriction of a Haar measure to a closed subgroup from being a Haar measure?

Let $\mu$ be a Haar measure on a locally compact Hausdorff topological group $G$, and let $H$ be a closed subgroup of $G$. If we restrict $\mu$ to the Borel sets of $G$ which are contained in $H$ ...
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54 views

Is a compact subset of a topological group G/N closed if G is hausdorff?

I just used this hypothesis when I was proving a theorem.But I was not sure if this hypothesis is correct.
3
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1answer
70 views

Unimodular groups

Let F be a non-archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series Fq((T)) over a finite field Fq (where q is a power of p). Is GL(n,F) ...
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35 views

From Folland Abstract Harmonic Analysis - Why is $\int f(x \xi) d\xi = \int f(x \xi^{-1}) \Delta_H(\xi^{-1}) d \xi$?

From Folland Abstract Harmonic Analysis - Why is $\int f(x \xi) d\xi = \int f(x \xi^{-1}) \Delta_H(\xi^{-1}) d \xi$? This is on page 57. Here is the notation: $H$ is a closed subgroup of a locally ...
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core-compact but not locally compact

A space $X$ is called core-compact if the set of all open set in $X, \mathcal{O}(X)$, is a continuous poset. It is known that every locally compact is core-compact. Here, a space $X$ is locally ...
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30 views

Show that a representative function on a profinite group factor through a surjection

Let $G$ be a compact group. A representative function $f\in\mathcal{C}(G,\mathbb{K})$ is a function such that $\dim\left(\operatorname{span}\left(Gf\right)\right)< \infty$. Remark that the ...
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1answer
34 views

Characters of a compact group with uniform positivity over $G$

Let $G$ be a compact group and let $\widehat{G}$ denote the set of all equivalence classes of irreducible representations of $G$. For each $\pi \in \widehat{G}$, we use $\chi_\pi$ to denote the ...
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1answer
112 views

When a sigma-finite space is a sigma-compact space?

$X$ is a topological space, $m$ is a $\sigma-$finite measure on $B(X)$, and what condition can make $X$ be a $\sigma-$compact space? This question is from topological groups (for me). Locally compact ...
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53 views

Any quotient of a compactly generated space is compactly generated

I found a note about compactly generated. This is the article http://www.math.uiuc.edu/~franklan/Math535_1205.pdf. I worry whether the proof of Proposition 2.4 is true. I not understand why the ...
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45 views

intersection about the second category

$G$ is a locally compact Hausdorff topological group, $A$ and $B$ are two Borel subsets of $G$, and $A$ and $B$ are both of the second category in $G$, then there exist an element $x\in G$, such that ...
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1answer
51 views

A reference for the Tannaka-Krein theorem

I am looking for a reference for the Tannaka-Krein theorem on compact groups. By the Tannaka-Krein theorem which is also called (classic) Tannaka duality (because of the quantum theory), I mean the ...
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1answer
88 views

Compact subgroups are contained in open compact subgroups in locally profinite groups

Let $G$ be a totally disconnected, Hausdorff, locally compact group. In the wikipedia page about these groups there is a claim that any compact subgroup of $G$ is contained in some compact open ...
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41 views

Two questions about increasing unions of compact subsets of a locally compact Hausdorff group.

I have two questions to ask related to my research. Question 1. Let $ G $ be a locally compact Hausdorff group. Is it possible that $ G $ is the union of a chain of compact subsets (ordered by ...
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81 views

locally compact Hausdorff

A space $X$ is called locally compact if every point of $X$ has a compact neighbourhood. I want to show that If $X$ is Hausdorff then $X$ is locally compact iff for every $x$ of $X$, every ...
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1answer
33 views

Sigma-compact Polish groups

I would like to see an example of a sigma-compact Polish group which is not locally compact. I know that e.g. $l^{\infty}$ is a topological group which is sigma-compact but not locally compact. But ...
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32 views

Eigenfunctions of the Fourier transform on locally compact abelian groups

The eigenfunction theory of the Fourier transform on $\Bbb R$ is well-understood. For example, the Hermite-Gauss functions are eigenfunctions with eigenvalues $i^n$; in fact, this comprises the ...
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1answer
71 views

What are the continuous automorphisms of $\Bbb T$?

I wanted to check my reasoning on this problem. From standard Pontrjagin duality arguments, it's not hard to see that the continuous homomorphisms of the torus (to itself) are nothing more than the ...
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1answer
195 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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39 views

closed subset of locally compact

A space $X$ is said locally compact if for any $x\in X$ and for any neighbourhood $U$ of $x$ there is a compact neighbourhood $V$ such that $V\subseteq U$. Does closed subset of locally compact is ...
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Convolution of matrix coefficients is also a matrix coefficients

I have a question about the convolution of matrix coefficients as follows: Let $G$ be a compact Lie group. A Map $f:G\rightarrow \mathbb{C}$ is called a matrix coefficient if there is a finite ...
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48 views

Intersection between a compact and a locally compact set

I'm trying to understand Rudin's proof of Pontryagin duality theorem, but I still haven't undersood an argument. (Fourier analysis on groups, p29) Let $G$ be a group and denote $\Gamma =\widehat{G}$ ...
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1answer
60 views

Is every closed subgroup of dual group an annihilator?

This is a naive question as I am new to topological group theory. Let $G$ be a Locally Compact Abelian group (LCA group). I know that to every closed subgroup $H$ of $G$ correspond a closed subgroup ...
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79 views

sequential continuity and countinuity

When we have two topological spaces, $\left(X, \tau_X\right)$ and $(Y, \tau_Y)$ it is easy to check that for $f: X \rightarrow Y$ continuity implies sequential continuity. I'm wondering what do we ...
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Stabilizer, Cosets, homeomorphism and Compact groups : proving things in The Structure of Compact Groups by Hofmann and Morris

I'm currently struggling trying to prove a few things in the book The Structure of Compact Groups by Hofmann and Morris. The first one would be Proposition 1.10.i (or E1.4) : If the topological ...