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 […]

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.

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 […]

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 […]

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 […]

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?

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 […]

Let $G$ be a locally compact group on which there exists a Haar measure, etc.. Now I am supposed to take such a metrisable $G$, and given the existence of some metric on $G$, prove that there exists a translation-invariant metric, i.e., a metric $d$ such that $d(x,y) = d(gx,gy)$ for all $x,y,g \in G$. […]

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 representative functions form a subalgebra of $\mathcal{C}(G,\mathbb{K})$. I’m following the book “The Structure of Compact Groups” by Hofmann&Morris on this subject. I would like to be able to show that that a representative function $f$ […]

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, or its integral operator. Of course, right invariant ones are as good.

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