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 \begin{equation} \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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Extension of a continuous and equivariant map to compactification

Given a locally compact (topological) group $G$ we denote by $C_b^{lu}(G)\subseteq C_b(G)$ the bounded continuous functions on $G$ such that $f\in C_b^{lu}(G)$ whenever the map $G\to C_b(G)$ given by $...
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Extending representations from a closed Lie subgroup - solution verification

Let $G$ be a compact Lie group and $H$ a closed Lie subgroup. Let $i : H \hookrightarrow G$ denote the inclusion. Then given any finite-dimensional representation $\rho : H \rightarrow \mathrm{GL}(V)$ ...
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Is $\mathcal{O}=\{\pi_{v,w}| \ (\pi,H)\in Rep_f^s(G),\ v,w\in H\}$ dense in $C(G)$? where $G$ is compact, $T_2$ group

Let $Rep_f^s(G)$ be the set of all finite dimensional, strongly continuous hilbert space representations of compact, $T_2$ group $G$. For $(\pi,H)\in Rep_f^s(G)$ and $v,w\in H$ define $\pi_{v,w}(g):=\...
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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 ...
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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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Does $(x,y)\mapsto \phi_x(y)$ vanish outside some compact set in $G\times H$?

Suppose that $G$ and $H$ are locally compact Hausdorff groups. EDIT: As Eric Wofsey pointed out in the comments; the group structures can probably be ignored for this question. Assume that for each $x\...
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Proof that the Dual of an LCA Topological Group is LC with the compact-open topology

Let G be a LCA Topological Group, I define it's dual group, $\Gamma$, by the group of continuous characters $$\gamma:G\to\mathbb{T}$$ where $\mathbb{T}$ is the complex unit circle. I want to equip ...
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Locally compact abelian group action vs graded algebras

This question is a follow-up to Elvorfirilmathredia's question Correspondence of grading and group actions. I wound up there while studying some $C^*$ quantum group stuff. I'm quite new in the topic, ...
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Continuous group action induces a continuous action on its function space

Let a topological group $G$ act (joint) continuously on a topological space $X$. Is it true that the induced action on $C_0(X)$ given by $s.f(x):=f(s^{-1}.x)$ for $s\in G, x \in X, f\in C_0(X)$ is ...
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$G$ acts transitively on a space $X$. If a function on $X$ is $G$-invariant up to measure zero, is it necessarily a constant (up to measure zero)?

Consider a locally compact Hausdorff $σ$-compact topological space $X$ and a locally compact Hausdorff $σ$-compact topological group $G$ acting continuously and transitively on $X$ such that there ...
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Citation for generalized poisson summation formula

I am wanting to cite the general Poisson summation formula for locally compact abelian groups (such as here). Does anyone know of a proper (preferably peer-reviewed) place where this is written down ...
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1 answer
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Compact set in a space with the product topology

I have zero clues on how to solve this: "Let be R with the discrete topology d, and R with the eucledian topology e, and X=R x R with the product topology t=d x e. X with the topology product t ...
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1 answer
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$K \subset \mathbb{R}$ is a compact set such that $ K=K'$ and $f:K \rightarrow \mathbb{R}$ is a function of class $C^1$, then $f$ is Lipschitzian.

Prove that if is a $\emptyset \ne K \subset \mathbb{R}$ is a compact set such that $ K=K'$ and $f:K \rightarrow \mathbb{R}$ is a function of class $C^1$, then $f$ is Lipschitzian. Can anyone help me ...
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3 votes
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Algebraic vs. continuous representations of topological groups

Let $G$ be a locally compact (Hausdorff) topological group and $\rho:G\to \operatorname{GL}(V)$ a complex continuous representation of $G$ in some locally convex topological vector space $V$. I'm ...
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Restriction-Induction Lemma for representations of standard split maximal torus of $GL_2(F)$

I am trying to understand the proof of the following lemma, which comes from section 9.3 of The Local Langlands Conjecture for GL(2). In this question, $T$ is the subgroup of $GL_2(F),$ $F$ a non-...
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When is $T_w := \{\mbox{diag}(i\theta) \mid \sum_i w_i \theta_i = 0\}$ the Lie algebra of a compact connected Lie subgroup of $SU(n,\mathbb C)$?

Let $SU(n,\mathbb C)$ be the special unitary group of size $n$. Note that the corresponding Lie algebra is given by $S_n$, the vector space of skew-symmetric traceless matrices of size $n$. Given a a ...
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$L^1$-density of the functions $F_{f,g}(s,t):=f(s^{-1}t)g(t)$ in $C_c(G\times G)\subset C_c(G,C_{0}(G))$

Suppose that $G$ is a locally compact group with Haar measure $\mu$. Note that we can view $C_c(G\times G)$ as a linear subspace of $C_c(G,C_{0}(G))$ by identifying the evaluations $F(s,t)$ and $F(s)(...
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2 votes
1 answer
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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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1 vote
2 answers
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If $\int_A f d \mu = \int_A g d\mu$ for continuous positive $f,g$ and a Radon measure $\mu$, is $f=g$ a.e.?

Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure on $X$. If $f,g: X \to (0, \infty)$ are continuous functions such that $$\int_A f d \mu = \int_A g d\mu$$ for all Borel subsets $...
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9 votes
1 answer
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Is a compactly supported function on a locally compact Hausdorff space uniformly continuous?

Consider the following fragment from Folland's book "A course in abstract harmonic analysis": Let $G$ be a locally compact Hausdorff group. I want to prove that the representation $$\pi: G \...
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2 votes
1 answer
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Every totally bounded metric space is locally compact? [closed]

I am aware of the facts that every totally bounded metric space is separable and a metric space is compact iff it is totally bounded and complete but I wanted to know, is every totally bounded metric ...
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1 answer
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Topology of classical compact groups?

Let $O(n)$, $U(n)$, $SO(n)$ and $SU(n)$ be the orthogonal, unitary, special orthogonal and special unitary group respectively. What are the topology of these groups? (I'm just a beginner to algebraic ...
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3 votes
0 answers
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Peter Weyl Theorem - Space of matrix coefficients dense in C(G)

So let G be a compact group. Peter-Weyl's theorem states that $$L^2(G)= \overline{\bigoplus_{\lambda \in \hat{G}}M_\lambda}. $$ Where $M_\lambda$ is the space generated by the coefficients of a ...
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1 answer
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Schur's Orthogonality relations

I am trying to calculate the same question that got asked a few years back. Identities related to Schur's orthogonality relations. But in the comments one guy gives a way to try and compute it ...
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Finite-dimensional irreducible representations and cyclicity

Let $\pi$ be a finite-dimensional irreducible unitary representation of a locally compact group $G$ on a complex Hilbert space $H$. Let $n = \dim H$. One may consider the induced unitary ...
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3 votes
2 answers
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Is Haar measure reflection invariant?

Let $G$ be a locally compact group and $\mu$ be a Haar measure on $G.$ Is $\mu$ necessarily reflection invariant i.e. can we always say that $\mu (E) = \mu (E^{-1})\ $? where $E \subseteq G$ is a ...
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1 answer
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Why is this function continuous in the proof of Folland's proposition 7.21

Let $X$ and $Y$ be locally compact Hausdorff spaces. Consider the following fragment from Folland's text "Real analysis", p226: I checked the errata and the $\overline{U}\times \overline{V}$...
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3 votes
0 answers
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Question on an inequality concerning integrals.

Let $G$ be a locally compact group and $f \in L^1(G).$ Define a linear map $L_f : L^2(G) \longrightarrow L^2(G)$ by $g \longmapsto f \ast g,\ $ $g \in L^2(G).$ Show that $L_f$ is bounded. Here we ...
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$\bigoplus_{i \in I} G_i$ is locally compact Hausdorff.

Let $\{G_i: i \in I\}$ be a family of topological groups. I have to prove that $\bigoplus_{i \in I} G_i$ is locally compact Hausdorff if and only if each $G_i$ is locally compact Hausdorff and $G_i=\{...
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1 answer
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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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15 votes
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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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5 votes
2 answers
156 views

Is the unitary group $U(H)$ with the strong operator topology locally compact?

Suppose that $H$ is a complex Hilbert space. Endow the unitary group $U(H)$ with the strong operator topology (SOT) - that is, $u_{i}\to u$ in $U(H)$ if and only if $u_{i}x\to ux$ in $H$ for all $x\in ...
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1 vote
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Smooth vectors in closed invariant subspace

Suppose $G$ is a Lie group, and $(\pi, E)$ is a continuous representation of $G$ on a Fréchet space $E$. Let $E^\infty$ be the set of smooth vectors of $E$, which is known to be dense in $E$. In ...
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5 votes
1 answer
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$\mathrm{Iso}(X)$ is locally compact if $X$ is locally compact

Let $(X, d)$ be a locally compact metric space. Define $\mathrm{Iso}(X)$, the isometry group of $X$, as the set of all surjective isometries of $X$. Is it true that $\mathrm{Iso}(X)$ is locally ...
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2 votes
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An example of a compact group which its identity connected component is nontrivial, abelian and of measure zero.

Denote by $G_0$ the identity connected component of a topological group $G$. Is there a compact group $G$ such that $G_0$ is abelian group, $Z(G)$ does not contain $G_0$, and $G_0$ is of (Haar) ...
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Folner nets for amenable groups and basis of neighborhoods at $e$

Let $G$ be an amenable locally compact group with identity $e$. Out of curiosity, can we choose a Folner net $(F_i)$ to be a basis of neighborhoods at $e$ for some classes of locally compact groups ?
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4 votes
2 answers
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Is unimodular stable under local isomorphisms?

Let $G$ and $H$ be locally compact groups. Suppose that $G$ and $H$ are locally isomorphic. If $G$ is unimodular, is it true that $H$ is unimodular ? Two topological groups $G$ and $H$ are said ...
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1 vote
1 answer
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Why $\text{SL}_2(\mathbb Z_p)$ is open in $\text{GL}_2(\mathbb Q_p)$?

Let $p$ be a prime number. Denote by $\mathbb Q_p$ and $\mathbb Z_p$ the field of $p$-adic numbers and the ring of $p$-adic integers. Why $\text{SL}_2(\mathbb Z_p)$ is open in $\text{GL}_2(\mathbb Q_p)...
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1 vote
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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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3 votes
1 answer
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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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1 answer
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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 ...
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4 votes
0 answers
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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 (...
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