0
votes
1answer
45 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 ...
2
votes
1answer
51 views

Can one visualise the dual groups to Cantor groups?

My question is very simple, and, probably an answer can be found in any harmonic analysis textbook, but it seems I have failed that task. It occurred to me that I don't understand the structure of the ...
2
votes
0answers
51 views

Abstract Fourier Analysis

I am trying to prove that given a locally compact abelian (Hausdorf) topological group, the characters on it parameterize the multiplicative linear functionals on the banach * algebra $L^1(G, d\mu)$ ...
0
votes
0answers
42 views

If a normal subgroup, N, contains a lattice why does G/N have finite measure?

Suppose $G$ is a locally compact Hausdorff topological group and suppose $H \leq N \leq G$ are closed subgroups with $N$ normal. Now suppose $G/H$ has a finite $G$-invariant Boreal measure (in the ...
5
votes
2answers
169 views

equation involving the integral of the modular function of a topological group

Let $G$ be a locally compact topological group and $H$ a closed subgroup. Choose a left Haar measure $d\zeta$ for $H$, and let $d\mu$ be any measure for $G$. Also let $f$ and $g$ be continuous ...
3
votes
0answers
49 views

Is Steinhaus theorem ever used in topological groups?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff ...
1
vote
0answers
42 views

right multiplication by elements of a discrete subgroup preserve left haar measure?

If $\Gamma$ is a discrete subgroup of a locally compact topological group, G, it is not necessarily the case that right multiplication on $G$ by elements of $\Gamma$ will preserve a left Haar ...
7
votes
2answers
205 views

Good book on topological group theory?

I'm looking for a good introduction to the theory of locally compact groups and their representations. It may assume the reader to be familiar with basic group theory, topology and measure theory.
6
votes
1answer
198 views

How does Pontryagin duality fit into the general cohomology theory framework?

Pontryagin duality implies the isomorphic relation of the function space $C(G)$ on a locally compact group $G$ to the function space on it's dual group $\hat G \overset{\sim}{=}\text{Hom}(G,T)$, ...
7
votes
1answer
276 views

Basics of Haar measure

Suppose $G$ is a locally compact group. Then $G$ has a left-invariant measure $dg$, say, which means that $$\int f (hg) dg = \int f(g) fg$$ for any test function integrable on $G$. The ...
4
votes
1answer
115 views

Consequences of Pontryagin Duality?

What are some interesting corollaries and consequences of the Pontryagin Duality theorem? My question can be taken as broadly as you'd like, even up to including any philosophy introduced specifically ...
7
votes
1answer
442 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 $\mathcal{L}^1(G, \mu)$. ...
2
votes
1answer
238 views

Example of a locally compact connected Abelian group with non-$\sigma$-finite measure

I look for an example of an Abelian locally compact topological group $G$ such that: $G$ is connected and Haar measure on $G$ is not $\sigma$-finite and $\{0\} \times G \subset \mathbf{R} \times G$ ...
0
votes
1answer
244 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 ...
3
votes
1answer
126 views

Restricted Direct Products in Koch's Number Theory

On p.353 of Number Theory: Algebraic Numbers and Functions by Helmut Koch, he considers a group $G$ which is the restricted direct product of the locally compact abelian groups $G_i$ with respect to ...