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 ...

learn more… | top users | synonyms

1
vote
1answer
24 views

Isomorphisms of LCA Groups

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 ...
0
votes
0answers
28 views

The definiton of a discrete group

Is there a definition of a discrete group different from the one given in following link: http://en.wikipedia.org/wiki/Discrete_group
4
votes
0answers
38 views

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 ...
1
vote
1answer
43 views

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 ...
1
vote
1answer
54 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 ...
3
votes
1answer
40 views

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 ...
0
votes
0answers
25 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 ...
1
vote
1answer
24 views

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 ...
0
votes
0answers
77 views

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) ...
0
votes
1answer
26 views

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 ...
2
votes
0answers
64 views

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 ...
3
votes
0answers
57 views

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 ...
2
votes
0answers
36 views

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 ...
0
votes
0answers
14 views

Divisible, locally compact abelian groups

Let $G$ be a divisible, locally compact abelian group such that $G$ is a direct sum of a torsion group and a torsion-free group. Let $H$ be an open subgroup of $G$. Can we deduce that $H$ is a direct ...
1
vote
1answer
22 views

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 ...
0
votes
1answer
18 views

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 $x$?
0
votes
0answers
9 views

Kernel of a representation and central projections

Let $G$ be a locally compact group, and $B(G)$ be the Fourier-Stieltjes algebra of $G$. I'm trying to see why the following is true: let $\pi$ be a representation of $G$, the the kernel of $\pi$ in ...
0
votes
1answer
31 views

Haar measure of point sets

Let $G$ be a locally compact group with Haar measure $\mu$ (left or right doesn't matter to me). I know that the Haar measure is positive on open sets. What can be said about the Haar measure on ...
1
vote
1answer
42 views

A sequence of neighbourhoods with decreasing Haar measure in a non-discrete group

Let $G$ be a non-discrete (LCH) group. How can one find a sequence $(V_n)$ of compact, symmetric neighbourhoods of the identity element $1$ such that $\mu(V_n)\to 0$, where $\mu$ denotes the Haar ...
1
vote
2answers
68 views

Totally disconnected orbit spaces

Let $X$ be a totally disconnected $G$-space, where $G$ is a locally compact Hausdorff group. Is the orbit space X/G also totally disconnected? The same question for locally compact, Hausdorff, ...
1
vote
0answers
33 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) ...
2
votes
1answer
58 views

Is the ring of integers of a local field an open subgroup?

I apologize if my question is a bit naive, but I don't have much experience in number theory and sometimes get very confused. Suppose $K$ is a non-archimedean local field (essentially a completion of ...
3
votes
2answers
53 views

Why do we want *unitary* representations of locally compact groups into $B(H)$?

This is related to a previous question of mine but I have a more philosophical issue with the material. Everywhere I have looked for representations of locally compact groups into $B(H)$, everyone ...
2
votes
1answer
39 views

Why is the image of $k^+$ dense in the character group?

Let $k$ be a local field, and consider the map $\phi: k^+ \hookrightarrow \widehat {k^+}$ given by $\eta \mapsto \eta X(\cdot)=X(\eta \cdot)$ where $X$ is a non-trivial character. Tate argues in his ...
1
vote
1answer
58 views

What is the cokernel of $\Bbb Q^{\text{disc}} \hookrightarrow \Bbb R$?

These two should be the standard examples for why Locally compact abelian groups are not an abelian categoty. The cokernel of any of these maps should is not a LCAG. $$\Bbb Q^{\text{disc}} ...
4
votes
0answers
75 views

Reduced $C^*$-algebra of a direct product of locally compact groups

Is it true that $$C^*_r(G_1\times G_2)=C^*_r(G_1)\otimes_{\min}C^*_r(G_2)$$ for locally compact groups $G_1$ and $G_2$? I have managed to prove that it holds for discrete groups (see below), but as ...
2
votes
0answers
25 views

Induced measure on dual group and kernel of fourier transform

Let $G$ be a locally compact Hausdorff abelian group (LCA). Then the Fourier transform gives a map from functions on G to functions on $\hat G$ $$f\mapsto \hat f(\xi) = \int_G f(x)\Psi(x,\xi) ...
2
votes
1answer
31 views

Integral of $|f|$ outside a compact set

Let $G$ be a locally compact group. Given $f\in L^1(G)$ and $\epsilon>0$, how to show that there is a compact set $K\subset G$ such that $\int_{G\setminus K}|f|<\epsilon$?
0
votes
1answer
71 views

Continuity of Modular Function

If $G$ is a locally compact group, there's a left invariant Borel measure on $G$, called Haar measure, which is unique up to multiplication by scalar. Denote it by $\mu$. For any $g\in G$, ...
1
vote
1answer
78 views

The dual of L^1(G) for a locally compact group G

I might be missing something, but most literature on topological groups and harmonic analysis that I've encountered mention that $L^\infty(G)$ can be naturally identified with the dual of $L^1(G)$ by ...
0
votes
1answer
17 views

Show $T:G\rightarrow L^1(G)$ by $y\mapsto f(y\cdot)$ is continuous for fixed $f\in L^1(G)$

Let $G$ be a locally compact group, and show $T_f:G\rightarrow L^1(G)$ by $y\mapsto f(y\cdot)$ is continuous for fixed $f\in L^1(G)$
0
votes
0answers
19 views

What is the relationship between the spaces $\mathscr K (G)$ and $L^2(G)$?

The context is that $G$ is a locally compact Hausdorff group, $\mathscr K (G)$ is the space of continuous compactly-supported functions $G \to \mathbb C$ equipped with the inner product $(f|g) = \int ...
2
votes
1answer
96 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", ...
2
votes
1answer
57 views

When does a left Haar measure on a locally compact group restrict to a left Haar measure on a locally compact subgroup?

Given a locally compact group which is not $\sigma$-compact, there exists a $\sigma$-compact subgroup $H$ of $G$ which is open and closed. A Remark in Folland's, A Course in Abstract Harmonic ...
1
vote
0answers
52 views

Construct a topological manifold which its open cover is locally finite but not globally

The whole question is like this: 1-4. Let M be a topological manifold, and let U be an open cover of M . (a) Assuming that each set in U intersects only finitely many others, show that U is locally ...
1
vote
0answers
104 views

Is a regular Borel measure on a locally compact space necessarily $\sigma$-finite?

I am trying to compile a proof of the uniqueness of Haar measure. Usually this is done by multiple-integral mumbo-jumbo, abusing left and right invariance of two potential measures and invoking ...
0
votes
1answer
27 views

Is group of rigid body motion compact?

I believe that group of rigid body motion is not compact. I mean all transformations in $R^3$ that preserve distance. But I need to know how to proof it? From where I should start to prove it?
3
votes
1answer
102 views

Is there a formula for the Haar measure on a product of groups?

Let $ (G_{n})_{n \in \mathbb{N}} $ be a sequence of locally compact topological groups with a corresponding sequence $ (\mu_{n})_{n \in \mathbb{N}} $ of Haar measures. Is there a way to construct a ...
1
vote
1answer
66 views

Convolution of $L^1(G)$ functions with elements of $M(G)$.

Let $G$ be a non-discrete locally compact group with left Haar measure $\mu$. There is an isometric embedding of $L^{1}(G)\to M(G), f\mapsto fd\mu$. Since $G$ is not discrete, the point-mass measure ...
2
votes
1answer
63 views

Why is $L^{1}(G)$ unital if and only if $G$ is discrete?

I've seen it stated in several sources and lecture notes for Abstract Harmonic Analysis that for a locally compact group $G$, $L^{1}(G)$ is unital if and only if $G$ is discrete. What about the ...
4
votes
1answer
46 views

Showing a measure on a locally compact group is left invariant

I am trying to verify that the measure $\frac{1}{|x|}dx$ is a Haar measure on $\mathbb{R}\backslash \{0\}$. For every open interval $(a_{n},b_{n})\subset \mathbb{R}$ not containing $0$, I have that ...
1
vote
1answer
125 views

is the vector space $\mathbb{R}^\mathbb{N}$ locally compact?

is the vector space $\mathbb{R}^\mathbb{N}$ locally compact? for example, let $x=(x_1,x_2,....)$ any point of $\mathbb{R}^\mathbb{N}$ and let $V=[x_1-\epsilon,x_1+\epsilon] \times ...
1
vote
1answer
109 views

Can a locally compact group with closed singleton be countable but not discrete?

Problem: Prove if a locally compact group $(G,*)$ contains a closed singleton then it must be either discrete or uncountable Proof Given: Assuming $G$ is countable we can write $G = \displaystyle ...
3
votes
1answer
34 views

Locally compact infinite dihedral group

Is there any topology other than discrete which can be given to an infinite dihedral group to make it locally compact topological group? If no, why is it so?
4
votes
0answers
156 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, ...
0
votes
1answer
128 views

How can this linear map be injective?

Studying the Peter-Weyl theorem, I've come across the following linear maps: $\theta_E: E'\otimes E \rightarrow$ Hom$(E,E)$ Where $ E'\otimes E$ denotes the tensor product of a finite-dimensional ...
1
vote
1answer
29 views

$SU_2(\mathbb{C})$ and the characters

i can prove that the irreducible characters $\chi_n$ of $SU_2(\mathbb{C})$ are equal to: $$\chi_n(e^{i\phi})=\frac{\sin((n+1)\phi)}{\sin(\phi)}$$ If i want to give the dimension of the representation ...
0
votes
1answer
28 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 ...
3
votes
0answers
80 views

Mapping $G$ into its group algebra as left multiplication. Continuous?

I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove: Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
1
vote
1answer
37 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 ...