Articles of locally compact groups

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 maps $z\mapsto z^n$ where $n\in\Bbb Z$. If $n\neq \pm 1$, these maps are clearly not bijective since we have multiple $n$th roots. […]

Compact group actions and automatic properness

I am currently re-reading a course on basic algebraic topology, and I am focussing on the parts that I feel I had very little understanding of. There is one exercise in the chapter devoted to groups acting on topological spaces (preceding the chapter on covering spaces) that I have spent several frustrating hours on, but […]

Steinhaus theorem in topological groups

Let $(G,\cdot)$ be a locally compact Abelian topological group with Haar measure. It is known that if $B$ is a measurable subset of $G$ of finite and positive Haar measure then $int(B \cdot B)\neq \emptyset$ (Hewitt, Ross, Abstract harmonic analysis I, Cor.20.17 p.296). My question is whether the interior of $B \cdot B$ has to […]

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 locally compact group $\mathbb{T} = \{\lambda\in\mathbb{C}: |\lambda| = 1\}$, which is not discrete because the arclength measure of a point on […]

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? Any notes or suggestions will be appreciated.

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 a character $\chi: H \rightarrow S^1,$ I need to construct an extension […]

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 means of the isomorphism $$f\mapsto\left(g\mapsto\int fg d\mu\right)$$ where $\mu$ is a fixed Haar measure. However, when dealing with measure spaces $(X,\mu)$ in general, a lot […]

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 compact set Is $\mu$-regular (i.e. outer and inner regular) 1) It can be shown as a consequence of the above […]

Surjective endomorphism preserves Haar measure

How to prove the following statement: Let $G$ be a compact topological group and let $m$ be the Haar measure on it. Let $\varphi$ be a continuous endomorphism of $G$ onto $G$, i.e., the map $\varphi$ is surjective. Then $\varphi$ preserves $m$. Is compact necessary, or is it still true for locally compact groups?

$\sigma$- compact clopen subgroup.

I am given $G$ locally compact group, and I want to show that there exists a clopen subgroup $H$ of $G$ that is $\sigma$-compact. So here’s what I did so far: for $e \in U$, where $U$ is a nbhd of the identity element we know that $x \in x\bar{U}$, and $x\bar{U}$ is compact in […]