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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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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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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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.
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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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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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Three questions from σ-compact spaces and topological groups [closed]

every locally compact subgroup of a Hausdorff group is closed. A Hausdorff and $σ-$compact space X is a Baire space if and only if the set of points at which is $X$ is locally compact is dense in ...
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30 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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67 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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28 views

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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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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34 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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41 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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69 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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24 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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59 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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16 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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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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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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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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35 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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58 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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34 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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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 ...
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39 views

Transpose in $ {SL}(2,\mathbb{R})$

Let $SL(2, \Bbb{R})$ denote the group of special invertible $2\times 2$ matrices over $\mathbb{R}$. As a locally compact group, the Haar measure of $SL(2, \Bbb{R})$ is computable through Iwasawa ...
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42 views

What's a Schwartz-Bruhat Function

Let $X$ be a locally compact abelian group and $f: X \rightarrow \mathbb{C}$ a continuous map. There are several definitions of what it means for $f$ to be a Schwartz-Bruhat function. If $X = ...
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Countable LCA groups

Is it true that a countable LCA group can only be discrete ? This question is related to a comment here : A theorem on LCA group - is the uncountability necessary?
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On the structure of non discrete locally compact topological (non-necessarily commutative) complete fields.

There is totally classic result about the structure of non discrete locally compact topological (non-necessarily commutative) fields $K$, whose proof uses the existence of the Haar measure on the ...
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28 views

Good reference for The Differntiable Slice Theorem

I am looking for a book that will give me a good proof of The Differentiable Slice Theorem - Suppose a compact Lie group $G$ acts smoothly on a manifold $M$. Then every orbit has a $G$-invarient ...
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37 views

Norm Inequality (Vinogradov Notation)

I'm going through a proof of differentiability of fourier series on the d-dimensional torus and while proving the following inequality: $$ ...
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1answer
34 views

Decompostion into countably many nowhere dense compact sets

Let $A$ be a meagre subset of a locally compact abelian Polish group $G$. Then $A$ can be written as a countable union of nowhere dense subsets of $G$. Is it always possible to write $A$ as a ...
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45 views

question regarding closure of symmetric neighborhood of e

I was going through the open mapping theorem (for topological groups) when I stumbled upon a topological property that I couldn't prove to myself. If I have a topological group, $G$ which is ...
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37 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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33 views

Compact groups are Moore groups

A locally compact group $G$ is said to be a Moore group if each irreducible continuous unitary representation of $G$ is finite dimensional. I'm trying to see why a compact group $G$ would be a Moore ...
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25 views

Automophism of G and Haar measure

Let $G$ be a locally compact group (written additively), $\lambda$ an automophism of $G$, and $\alpha$ a Haar measure in $G$. As the Haar measure is unique up to factor constant, $\lambda$ transform ...
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How can we topologize the character group $\Gamma$ of a locally compact abelian group $G$, such that $\Gamma$ becomes a LCA group?

How can we topologize the character group $\Gamma$ of a locally compact abelian group $G$, such that $\Gamma$ itself becomes a LCA group ? I would really really appreciate if I can get a step by step ...
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measure of open set with measure Haar

By a Haar measure on a locall compact group (Hausdorff) we mean a positive measure $\mu$ (contains the borel set's) such that The measure $\mu$ is left invariant The measure μ is finite on every ...
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51 views

Error in Cariolaro's Unified Signal Theory

From what I understand, in the category $\mathsf {LCA}$ of lca groups, isomorphisms should respect both topology and group structure, hence they are continuous homomorphisms. I'm trying to learn ...
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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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Connected Pontryagin dual

The dual group to a compact abelian group is discrete so in particular very much disconnected. I was trying to invent an example of a connected locally compact abelian group with connected dual which ...
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68 views

Why is the character group defined as $\mathsf{Hom}(G,\mathbb T)$, i.e why is the codomain specifically $\mathbb T$?

In the paper Category Theory Applied to Pontryagin Duality by Roeder, the character group of an lca group is defined as the topological (under the compact-open topology) abelian group of continuous ...
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question on subgroup of compact group

Suppose $G$ is a compact metrizable abelian group, is it true that $G$ has no finite index subgroups iff $G$ is connected? Any reference or help is appreciated. Thanks in advance! Here are my ...
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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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Bounded approximate identities in one-sided ideals of $L_1(G)$

Let $G$ be a locally compact group and consider its $L_1$-group algebra, that is, $L_1(G)$. It is easily seen that $L_1(G)$ has a two-sided bounded approximate identity: just use a basis of compact ...
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Compact operators on $L^2(G)$ as a reduced cross product of $C_0(G)$ and $G$.

If any of the terminology is unclear then please don't hesitate to point it out. My question is: is it true that when $G$ is a locally compact second countable group then: \begin{equation*} C_0(G) ...
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Module function on automorphisms of discrete locally compact group

Let $G$ be a discrete locally compact group and let $\alpha: G \to G$ be an automorphism. Show that the module $\rm{mod}_{G}(\alpha)$ is 1. In the case of a locally compact field $k$ and ...
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Integration with values in a $C^*$-algebra

My question is quite specific to locally compact groups but I'm sure it can be generalised to locally compact Hausdorff spaces with a Borel measure. Let $G$ be a locally compact group and fix a Haar ...
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Building up the Fourier inversion theorem on locally compact abelian groups

I'm reading through Folland's Abstract Harmonic Analysis and I've come to a bit of a road block with some machinery developed for the Fourier inversion theorem. We know that the Fourier transform of a ...
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Locally compact group

I want/need to learn about locally compact groups in rigorous way. Mainly to completely understand Tate's thesis/Weil's basic number theory. Could you recommend a text or lecture note which would ...
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Discontinuous characters on locally compact abelian groups

I'm doing some reading out of Rudin's Fourier Analysis on Groups and Folland's Abstract Harmonic Analysis for research. Particularly I am on characters and dual groups and I've noticed that neither ...
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Containing a net with a compact set.

Let $G$ be a locally compact group and let $(x_{\alpha})$ be a convergent net, say to $x$, in $G$. Is it possible to construct a compact subset $K$ of $G$ which contains each $x_{\alpha}$ and $x$?