Questions tagged [locally-compact-groups]

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 relevant other tags whenever appropriate in order to reflect the main intentions of the question in the tags.

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Discrete abelian group of finite rank

By definition of rank, a torsion-free abelian group of rank $k$ is isomorphic to a subgroup of $\mathbb{Q}^k$. So it is true that a discrete torsion-free abelian group of rank $k$ is isomorphic to a ...
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When does Inverse Fourier transform look close to a positive definite function?

Let $G$ be a commutative locally compact group, and $\hat{G}$ be its dual group, consisting of all continuous characters (continuous homomorphisms from $G$ to the circle group $\mathbb{T}$) . I can ...
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Exercise 10 in section 11, chapter V, Conway's functional analysis.

On page 163, section 11, chapter V, Conway's functional analysis. Let $G$ be a locally compact group and $f\in\mathrm{C}_b(G)$. We say $f$ is almost periodic if \mathcal{O}_f\,\colon\...
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Let $G$ be locally compact abelian, $T_2$ and $f\in L^1(G)$. Define $\mu(A)=\int_A f(x)\ dx$. Prove $\lVert \mu\rVert=\lVert f\rVert_1$

Here $\mu$ becomes a complex measure and $\lVert \mu\rVert =|\mu|(G)$ is the total variation norm of $\mu$. We have to show $|\mu|(G)=\int\limits_G |f(x)|\ dx$ Let $\{A_n\}$ be a partition of $G$. ...
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Can a complex Radon measure be approximated by compactly supported Radon measures?

Let $G$ be an (abelian) locally compact Hausdorff group. Consider the following fragment from Folland's text "A course in abstract harmonic analysis" (second edition, p102). Why is the ...
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Why is $\phi_\mu(x)= \int_{\widehat{G}} \xi(x)d\mu(\xi)$ a continuous map?

Let $G$ be a locally compact Hausdorff abelian group. Let $\widehat{G}$ be its dual group, consisting of the unitary characters $G \to \mathbb{T}$. If $\mu \in M(\widehat{G})$ (= complex Radon ...
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Injectivity of divisible locally compact abelian groups

Are divisible locally compact abelian groups injective as objects of the quasi-abelian category of locally compact abelian groups ? At the very least, if $D$ is a divisible locally compact abelian ...
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Inner regular Haar measure

If $X$ is a locally compact Hausdorff space (l.c.H. space) then the Riesz representation theorem says that positive linear functionals on $C_c(X)$ are given by Borel measures on $X$. Unfortunately ...
1 vote
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Let $G=SO(3)$ and $\theta:G\to G$ defined by $\theta(g)=JgJ$ where $J=\text{diag}(-1,1,1)$. Prove that, $\theta(g)\in Kg^{-1}K$ where $K=SO(2)$

Here $K=\left\{\begin{pmatrix}1 & 0\\ 0 & T_1\end{pmatrix}:\ T_1\in SO(2)\right\}$. $K$ is a compact subgroup of $G$. The above problem is part of the proof of $(G,K)$ is Gelfand pair. I've ...
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Constructing unitary representations using quasi-invariant measures

Consider the following fragment (p.74) from Folland's book A Course in Abstract Harmonic Analysis (2e): My question is simple: What is the right handed situation of the above? I.e. suppose that $G$ ...
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The action of a compact group is continuous with respect to the product topology?

I'm reading the following paper https://arxiv.org/pdf/1804.10306.pdf in particular proposition (2.1) and trying to understand some facts I'm missing about compact groups theory, If I have a compact ...
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When are the Figà-Talamanca Herz algebras distinct from the Fourier algebra?

It is known that if $G$ is an amenable locally compact group and $1 < p < \infty$ then the Fourier algebra $A(G)$ is contained in the Figà-Talamanca Herz algebra $A_p(G)$ and that the $A(G)$ ...
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Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$.

Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly supported functions $C_c(G)$ are dense in $L^2(G)$. In the book "...
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Anticommutation of Convolution Products on Trace Class Operators of Quantum Groups

Edit: This question has been posted to MathOverflow. Let $\mathbb{G}$ be a locally compact quantum group and let $W$ and $V$ be the left and right fundamental unitaries, i.e., they implement the ...
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There is a unique Borel measure on $(0,\infty)$ with scaling invariance and $\mu([1,e])=1$.

Suppose that $\mu$ is a Borel measure on $(0,\infty)$ with $\mu([1,e])=1$ and for all $c>0$ and all $A$ Borel measurable, we have $\mu(cA)=\mu(A)$. Show that there exists a unique $\mu$ with these ...
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$f\in L^1(\mu) ~\text{and}~\forall g\in L^1(\mu)~\int gf ~ \mathrm{d}\mu=0 \implies f=0$ a.e.
My Definitions. Let $G$ be a locally compact Hausdorff group and $\mu$ be a Haar measure on $G$. We have defined "Haar" measure as follows: (Haar measure) It's a nonzero left invariant ...
When is the group $C^*$-algebra of a locally compact group an AF-algebra?
It is known that the group $C^*$-algebra of a compact group is an AF-algebra. I want to know if given a non-compact locally compact group $G$, does there exist conditions on $G$ which imply that the (...