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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Is the set of integers locally compact

I know $\mathbb{Q}$ is not locally compact as all compact subsets of $\mathbb{Q}$ have empty interior. Can I use the same argument to show that $\mathbb{Z}$ is not locally compact? All spaces are with ...
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Why does the dual space consisting of unitary equivalence classes of irreducible representations of compact $G$ not form a group?

I am trying to understand unitary representations of compact groups $G$. Equip the class of all unitary irreducible representations of $G$ with the usual equivalence relation, unitary equivalence. We ...
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Folland: Find Irreducible Representations of $SU(2)$ and decomposition of $L^2$ via Fourier analysis on compact groups.

I am working through the following textbook: Folland, Gerald B., A course in abstract harmonic analysis, Textbooks in Mathematics. Boca Raton, FL: CRC Press (ISBN 978-1-4987-2713-6/hbk; 978-1-4987-...
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Can someone provide intuition for Alfsen's proof of the existence and uniqueness of the Haar measure on locally compact groups?

I am trying to understand Alfsen's proof of the existence and uniqueness of the Haar measure on locally compact groups: Alfsen, Erik M., A simplified constructive proof of the existence and uniqueness ...
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What is the need of mentioning continuity in the proof of Frobenius reciprocity?

I can understand why the image of $\alpha\in \operatorname{Hom}_K(U,\operatorname{ind}_M^K(V_\sigma))$ lies in the subspace of continuous functions of $\operatorname{ind}_M^K(V_\sigma)$, and then the ...
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$f \in C_c(G) \implies \tilde f \in C_c(G)$ where $ \tilde f(x):=\int_K \int_Kf(k_1xk_2) dk_1dk_2$

Let $G$ be a locally compact, Hausdorff, non-abelian group and $K$ a compact subgroup of $G$. Let $f \in C_c(G)$ then prove that $\tilde f \in C_c(G)$ where $ \tilde f(x):=\int_K \int_Kf(k_1xk_2) ...
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How does Riesz representation theorem give us the left invariance of the Radon measure from a left invariant positive linear functional on $C_c(G)$?

I am trying to understand the end of the (canonical) proof of the existence of Haar measure on a locally compact group by Riesz representation theorem. Let $G$ be a locally compact group and $C_c(G)$ ...
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Pontryagin Duality and Injective Direct Limits

Let $(G_n)_{n \in \Bbb{N}}$ be an injective direct limit of discrete groups. How does pontryagin duality play with injective direct limits? More precisely, what is $\widehat{\lim G_n}$ isomorphic to? ...
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Adelic rings - reference request

I have been trying to find good resources on adelic rings. I am particularly interested in the context of homological algebra (i.e. $\operatorname{Ext} (\mathbb{Q}, \mathbb{Z}) $) and how they ...
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Small question on the universal group $C^*$ of a locally compact group

If $G$ is a discrete group, we can consider a family $(\pi_j)_{j \in I}$ of unitary representations of $G$ in which every equivalence class of a cyclic unitary representation of $G$ has an equivalent ...
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Prove that every compact subspace of a locally compact space $X$ has a compact neighbourhood

Let $X$ be locally compact space.Prove that every compact subspace has a compact neighbourhood. We should prove that for any $K$ compact there exists $H\subset X$ compact such that $K \subset H^{\...
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When does the conjugation action preserve Haar measure?

Let $H\trianglelefteq G$ be a normal closed inclusion of locally compact groups. Then $G$ acts by conjugation on $H$, thus on the one-dimensional space of Haar measures on $H$. For this action to be ...
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When is a locally Borel set Borel?

Let $X$ be a LCH space. Call $E\subseteq X$ locally Borel if $E\cap K$ is Borel for all compact $K\subseteq X$. Evidently, locally Borel sets form a $\sigma$-algebra and every Borel set is locally ...
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Pontryagin dual of the multiplicative group of a local field.

Let $K$ be a local non-Archimedian field. Let $K^{\times}$ be the group of invertible elements of $K$. Is there an explicit description of the Pontryagin dual of $K^{\times}$?
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Induced representations: space of continuous functions on $G$ to a Hilbert space

EDIT: Answered in MathOverflow @https://mathoverflow.net/questions/382324/induced-representations-space-of-continuous-functions-on-g-to-a-hilbert-space Let $G$ be a locally compact group, $H$ a ...
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What is known about the group of accumulation points of $(x^n)_n$ in a compact semigroup?

Let $S$ be a compact Hausdorff semigroup. Let $x ∈ S$ and let $A$ be the set of accumulation points of $(x^n)_{n>0}$. Here is a proof that $A$ is a group. I will write $ℕ$ for the set of strictly ...
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Topological groups and cardinality of their commutator group (example request)

What would be examples of non abelian, non discrete, second countable compact topological groups with a finite or at most countable commutator (derived) subgroup ?
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Compactness of units in ring of integers

Question : Let $\nu$ be a non-archimedean valuation on a number field $k$. Is $\mathcal{O}_{\nu}^{*}$ ( the set of units in ring of integers) compact under subspace topology of $k_{\nu}$ (completion ...
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Why isn't the Fourier transform self-inverse?

I'm reading the Wikipedia article on Pontryagin duality. (This is a fuzzy question about intuition for the difference between the forward and inverse Fourier transforms in this context.) It defines ...
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Application of Peter-Weyl theorem: groups with no small-subgroups admit a faithful finite-dimensional representation

My version of the Peter-Weyl theorem says that if $G$ is a compact group, then the matrix coefficients of $G$ are uniformly dense in $C(G)$. Consequently, the matrix coefficients are also dense in $L^...
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Is $f*\mu(x)=\int f(xy)dy$?

I am only recently familiarizing myself with basic harmonic analysis so pardon me if my question seems odd. I have a locally compact abelian topological group $G$. For $f:G\rightarrow \mathbb{C}$ I ...
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Some topological properties of “countable lines with one origin”

Let the countable lines with one origin to be a quotient space $CL = ([0, \infty) \times \mathbb N) / \sim$, where $[0, \infty) \times \mathbb N$ has a subspace topology of $\mathbb R^2$ and $0 \times ...
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Definition of finite dimensional representation of a compact group

Consider the following fragment: One asks that the map $\pi: G \to GL(V)$ is continuous. What does this mean? If $V$ is finite-dimensional, then we can view $GL(V)$ as matrices and then I guess it ...
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Example of a locally compact but not locally connected group

Does someone know any example of a locally compact but not locally connected group? Thank you
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Countable basis of compact open subgroups in locally profinite group.

I am currently reading through a proof of the existence of a right haar measure for locally profinite groups. We have a locally profinite group $G$ with the assumption that for all compact open ...
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Product of two left Haar measure

$\textbf{Theorem:}$ Let $A,B$ two locally compact groups and $G=A\times B$ their product. Let $\mu_A$ and $\mu_B$ a left Haar measures. Then $\mu_A \times \mu_B$ is a left Haar measure of $G$. $\...
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Haar measure on $\operatorname{GL}_n(\mathbb{R})_{+}$

Let's consider the group $$\operatorname{GL}_n(\mathbb{R})_{+} = \left\{ M \in M_n(\mathbb{R}) \mid \det(M) > 0\right\}$$ We identify this set as a open subset of $\mathbb{R}^{n^2}$. It is known ...
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Show $x \mapsto \int_G f(yx) \mu(dy)$ is continuous

This is exercise 11.2 in Folland's book: Let $\mu$ be a Radon measure on the locally compact group $G$ and $f \in C_c(G)$. Prove that $$x \mapsto \int_G f(yx) \mu(dy)$$ is continous. Before ...
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Square-integrability in lemma 4.30 of Folland's “A Course in Abstract Harmonic Analysis”

Update: I decided to post the question on MathOverflow here. In lemma 4.30 of Folland's "A Course in Abstract Harmonic Analysis" (Second Edition) one needs to show the square-integrability ...
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Terminology: Smooth Representations of Locally Profinite Groups.

Let $G$ be a locally profinite group. A smooth representation is a complex representation ($V,\rho$) of $G$ such that the stabiliser of any $v \in V$ is open. One can show that (as $\text{GL}_n(\...
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Weil's completion of locally totally bounded groups to locally compact groups

Let $G$ be a topological group. Say that $A\subseteq G$ is totally bounded if for any neighborhood of $e$, $U$, there exists a finite subset $F\subseteq G$ such that $A\subseteq FU$. $G$ is locally ...
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Kernel of restriction to dense subgroup is again dense

Let $X$ and $Y$ be topological compact abelian groups, and let $A$ and $B$ be dense subgroups of $X$ and $Y$ respectively. Let $\varphi\colon X \to Y$ be a continuous homomorphism such that $\varphi(A)...
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Unimodularity: How are these notions related?

1. Definitions We call a Hopf algebra $H$ unimodular if the space of left integrals $I_l(H)$ is equal to the space of right integrals $I_r(H)$. We call a square integer matrix $M$ unimodular if $det(...
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How to prove that $\mathbb{C}_p$ is not locally compact? [duplicate]

Let $\mathbb{C}_p$ be the $p$-adic complex numbers, i.e. the completion of an algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. How can one show that $\mathbb{C}_p$ is not locally compact?...
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Approximation of $L^p$ functions by convolution of approximate identities

We use the notation $\mathbb{K}$ to mean either the real number field $\mathbb{R}$ or the complex number field $\mathbb{C}$. Let $X$ be a locally compact group and let $\lambda$ denote the unique Haar ...
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Constant Function on the Support of Another Function

Let $G$ be a locally compact group and let $f_1,f_2 \in C_c(G)$ be nonnegative functions. I am working through a proof of a theorem and it in the author claims that we can choose $f'$ such that $f' \...
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Positive Functions in $C_c(G)$

Here is an excerpt from the book Principles of Harmonic Analysis by Deitmar and Echterhoff: Write $C_c^+(G)$ for the set of all positive functions $f \in C_c(G)$ [Note that $G$ is a locally compact ...
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Request: References and/or guidelines to assess the compactness of $D_{\infty h}$ [closed]

Can the community suggest "good" references for learning how to assess the compactness or lack of thereof in this particular case? Otherwise, I may need guidelines to approach/solve the ...
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Defining p-adic numbers via a formal completion.

Consider following fragment of the definition of p-adic numbers in "A course in abstract harmonic analysis" by Folland: So we have that $+: \Bbb{Q} \times \Bbb{Q} \to \Bbb{Q}$ is continuous ...
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Support of Measurable Function on Locally Compact Group

I am trying to work through the proof of the following theorem: Let $f$ be a measurable function on the locally compact group $G$, which is integrable with respect to a Haar measure. Then the support ...
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Splitting of homomorphism onto Prüfer group

Let $G$ be a totally disconnected locally compact abelian group. Let $U \leq G$ be open and such that $G/U \cong \mathbb{Z}(p^\infty)$, the Prüfer $p$-group. In general $U$ is not necessarily a direct ...
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For any $n \in \Bbb N$, for any representation $\phi:SL_2(\Bbb R) \to U(n)$ we must have $\phi \begin{pmatrix} 0 &1 \\ -1 &0 \end{pmatrix}= I_{n}$

Actually this is a continuation of this question, but am asking it separately as it deserves separate discussion as an independent problem. Thanks to comments by Exodd and an answer by Tsemo Aristide,...
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Proving $SL_2(\Bbb R)$ has no finite dimensional, non-trivial unitary representations using these hints

Show that $G=SL_2(\Bbb R)$ has no finite dimensional unitary representations except the trivial one. Let $A(t)=\begin{pmatrix}1 &t\\0 &1\end{pmatrix}, \forall t \in \Bbb R$ . Steps to follow : ...
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Why are all compact Lie groups matrix Lie groups?

I've seen it said that this is a consequence of the Peter-Weyl theorem (here), but I don't know how to do this and have been unsuccessful in finding a reference.
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Simple proof of the equality $\lim_{n \to \infty} \lVert f^{*n} \rVert_1^{1/n} = \lVert \hat{f} \rVert_\infty$

Let $G$ be a locally compact Abelian topological group and $\widehat{G}$ its Pontryagin dual, i.e. the group of all continuous characters of $G$. By $L^1(G)$ I will denote the space of all complex-...
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1answer
29 views

Intersection of lattice and closed subgroup

Let $G$ be a locally compact group, $\Gamma$ a (uniform) lattice in $G$. Let $H \leq G$ be a closed subgroup of $G$. Is $H \cap \Gamma$ a (uniform) lattice in $H$? In the discrete case, this is easy ...
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Haar Measures of $U(d)$ and $O(d)$

I'm trying to compute the Haar measures of finite unitary group $U(d)$ and finite orthogonal group $O(d)$. I've shown that both are compact groups. Most of computings I've seen base on Peter-Weyl ...
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1answer
37 views

Reference for $p$-adic Haar integral

I have stumbled upon the notion of a $K$-valued Haar integral on a locally compact group, where $K$ is a non-Archimedean field, as well as the $K$-valued modular function, in an article of Schikhof. ...
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Defining left Haar measure on locally compact group

Let $G$ is a locally compact group that is homeomorphic to an open subset (say $U$) of $\Bbb R^d$ ,and let $\varphi$ be a homeomorphism of $G$ onto $U$. Show that if for each $a$ in $G$ the function $...
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Isometrically embedding $L^1(H)$ into $L^1(G)$

Let $G$ be a locally compact group and $H$ a closed subgroup of $G$. Denote $\mu_G$ as Haar measure on $G$ and $\mu_H$ as Haar measure on $H$. Is it possible to embed $L^1(H)$ into $L^1(G)$? By Radon-...

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