All Questions
Tagged with topological-groups harmonic-analysis
68 questions
21
votes
3
answers
1k
views
Is a reductive adelic group a Type I group?
I foresee that to experts of automorphic forms this question will sound unimportant or useless or even not worthy of an answer; but none of these are going to stop me from asking it!
The question is ...
- 955
20
votes
0
answers
333
views
Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$
Remark: I cross-posted this question on MSE and added a bounty to it.
Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...
- 321
16
votes
1
answer
1k
views
A possible mistake in Walter Rudin, "Fourier analysis on groups"
I have the following lemma 4.2.4 on page 80 in the book (we have locally compact abelian topological groups $G_1, G_2$ and their duals $\Gamma_1, \Gamma_2$):
Suppose $E$ is a coset in $\Gamma_2$ ...
- 556
15
votes
3
answers
3k
views
Why is the dual of a torus the same as its fundamental group?
The set of continuous homomorphisms from a torus ${\mathbb T}^n = ({\mathbb R}/{\mathbb Z})^n \to {\mathbb R}/{\mathbb Z}$ can be identified with ${\mathbb Z}^n$ if we assign to each $k = (k_1, \ldots ...
- 2,243
15
votes
1
answer
498
views
For what LCH groups is the Haar measure $\mu(U x U)$ bounded?
Let $G$ be a locally compact Hausdorff (LCH) topological group with left Haar measure $\mu$. Given a compact unit neighborhood $U$, consider the function
$$
\Phi: \quad G \to (0,\infty), \quad x \...
- 734
13
votes
3
answers
2k
views
Weil's book L'intégration dans les groupes topologiques et ses applications
The book L'intégration dans les groupes topologiques et ses applications published by André Weil in 1940 is regarded as one of the classical references for harmonic analysis on topological groups.
...
user92170
13
votes
1
answer
852
views
Mistake on article about Bohr compactification?
$\DeclareMathOperator\b{b}\newcommand\B{{\operatorname B}}$I wish to get help understanding the content of two theorems of [Iva] that seem mutually contradictory. First some context. Let $\b(\mathbb{R}...
- 193
11
votes
1
answer
5k
views
Are there extremally disconnected groups?
A Hausdorff space is called extremally disconnected or extreme, if for every open set $U$ the closure $\overline U$ is open, too. The question, whether there are extremally disconnected topological ...
user473423
11
votes
1
answer
2k
views
Understanding Bruhat's notion of Schwartz function
I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwartz-Bruhat space $\mathscr{...
10
votes
2
answers
594
views
Existence of a strongly continuous topologically irreducible representation of a compact group on an infinite dimensional Banach space?
Does there exists a triple $(G, X, \pi)$, where $G$ is a compact group, $X$ an infinite dimensional Banach space over $\mathbf{C}$, and $\pi : G \to B(X)$ a strongly continuous representation of $G$, ...
- 960
8
votes
0
answers
167
views
A basis of the Banach space $L^p(\mathbb T^\omega)$ consisting of characters
Problem: For $1<p<\infty$, $p\ne 2$, has the complex Banach space $L^p(\mathbb T^\omega)$ got a Schauder basis consisting of characters of the compact topological group $\mathbb T^\omega$?
(...
- 4,535
7
votes
2
answers
1k
views
Bohr compactification as a topological compactification
Let $G$ be a locally compact Hausdorff group. Denote its Bohr compactification by $bG$.
Despite group structure, $G$ has several (Hausdorff) compactifications that, in a sense, the smallest one is ...
- 747
7
votes
0
answers
291
views
Degree of automorphic forms, SL(3,Z), and the elliptic Gamma function
In this article, the authors interpret a certain special function, the elliptic Gamma function, defined as
$$
\Gamma(z,\tau,\sigma)=\prod_{j,k=0}^\infty\frac{1-e^{2\pi i((j+1)\tau+(k+1)\sigma-z)}}{1-...
- 91
6
votes
1
answer
859
views
How many Fourier coefficients vanish?
Let $G$ be a compact abelian connected metric group with Haar measure $\mu$ and let $f\colon G\to S^1$(=unit circle in $\mathbb{C}$) be a continuous function (not necessarily a group homomorphism) ...
- 3,031
6
votes
1
answer
558
views
Inclusion of lattices and fundamental domains
Let $G$ be a locally compact abelian group. A lattice in $G$ is a discrete subgroup $\Lambda$ such that the quotient $G / \Lambda$ is compact. A Borel fundamental domain of a lattice $\Lambda$ in $G$ ...
user92170
6
votes
1
answer
1k
views
Classification of compact connected abelian groups
It is known that torsion-free compact abelian groups are exactly the product of the maximal solenoid group $\Sigma_{(2,3,\cdots)}$ (which is the Pontryagin dual of the additive group $\mathbb{Q}$ of ...
- 1,586
6
votes
1
answer
273
views
A property of rapid sequences of natural numbers
$\newcommand{\IR}{\mathbb R}$
$\newcommand{\IT}{\mathbb T}$
$\newcommand{\w}{\omega}$
$\newcommand{\e}{\varepsilon}$
Taras Banakh and me proceed a long quest answering a question of ougao at ...
- 5,409
6
votes
1
answer
377
views
Pontriagin reflexivity of the character group
For an Abelian topological group $G$ by $G^{\wedge}$ we denote the Pontryagin dual of $G$, i.e. the group of continuous homomorphisms $G\to\mathbb T:=\{z\in\mathbb C:|z|=1\}$. The group $G^{\wedge}$ ...
- 4,535
6
votes
0
answers
76
views
About path-connected components of the Bohr compactification of $\mathbb{R}^d$
Let ${\rm b}(\mathbb{R}^d)$ denote the Bohr compactification of $\mathbb{R}^d$, with $d\in\mathbb{N}$. This is the Pontryagin dual of the group $\mathbb{R}^d_d$, corresponding to $\mathbb{R}^d$ with ...
- 193
6
votes
0
answers
92
views
Does every compact abelian group contain a Kronecker set generating a dense subgroup?
Let $G$ be a compact metrizable abelian group with infinite exponent.
Let $S^1 = \left\{z \in \mathbb{C} : |z| = 1 \right\}$. A set $K \subset G$ is a Kronecker set if, for every continuous function $...
- 198
5
votes
2
answers
418
views
Is the left-regular representation of a locally compact group a homeomorphism onto its image?
Consider the left-regular representation $\lambda : G \to B(L^2(G))$, $\lambda_g f(h) = f(g^{-1}h)$, for a locally compact group.
It is well-known that this is a unitary faithful and strongly-...
- 769
5
votes
1
answer
611
views
What are the LCA groups that are the Pontryagin dual of a locally profinite abelian group?
For certain subcategories of LCA groups, we have nice descriptions of the dual category under Pontryagin duality (all groups are implicitly assumed to be abelian):
finite groups $\leftrightarrow$ ...
- 804
5
votes
1
answer
165
views
Is norm-continuous representation factored through a Lie quotient group?
I asked this 11 days ago at MSE, but there was no answer, I hope people here could help.
Let $G$ be a locally compact group, and $X$ a Hilbert space. A unitary representation $\varphi:G\to B(X)$ is ...
- 7,426
5
votes
1
answer
304
views
Tensoring with an induced representation: proof question
Let $G$ be a locally compact Hausdorff group and $H$ a closed subgroup of $G$. If $\sigma: H \to B(\mathcal{K}_\sigma)$ is a unitary representation of $G$, we can associate an "induced ...
- 175
5
votes
0
answers
132
views
Is $\mathbf{C}_p(X)$ self-dual?
Let $X$ be a set. Consider $\mathbf{Q}_p$ and $\mathbf{Z}_p$ as the $p$-adic numbers and $p$-adic integers, respectively. For any finite subset $F \subseteq X$, one can construct the topological ...
- 1,485
5
votes
0
answers
194
views
Haar mesure on $\mathrm{GL}_{d}(F)$
$\DeclareMathOperator\GL{GL}$Let $F$ be a $\mathfrak{p}$-adic field and $\mathscr{O}_{F}$ its valuation ring. For any measurable subset of $M_{d}(F)$ such as
$$
A=
\left( \begin{array}{ccc}
a_{11}+t^{\...
- 479
5
votes
0
answers
143
views
Two cardinal characteristics of the continuum, related to the Bohr topology on integers
For a subset $A\subseteq\mathbb T$ of the unit circle $\mathbb T=\{z\in\mathbb C:|z|=1\}$, let $\tau_A$ be the smallest topology on the additive group of integers $\mathbb Z$ such that for every $z\in ...
- 41.9k
5
votes
0
answers
119
views
Characterizing Herz-Schur multipliers using coefficient functions of uniformly bounded representations
Let $G$ be a group and let $c > 1$ be a constant. We denote by $B_c(G)$ the space of all coefficients of the representations of $G$ which are uniformly bounded by $c$; more precisely, a function $f:...
- 299
4
votes
2
answers
227
views
Root of positive function in Fourier algebra
Let $G$ be a locally compact group, let $A(G)$ be the Fourier algebra of $G$. We think of $A(G)$ as a subalgebra of $C_0(G)$.
Question 1: Let $f\in A(G)$ be a function that is pointwise positive. ...
- 3,497
4
votes
1
answer
221
views
Fourier multipliers and transference on cyclic groups
It seems to be a commonplace in harmonic analysis that if some operator (say, Fourier multiplier) is bounded on $L^p(\mathbb{R}^n)$ then by transference the similar operator is also bounded on $L^p(\...
4
votes
1
answer
304
views
Finite covolume of uniform lattice in quotient group
Let $G$ be a locally compact group, let $N \leq G$ be a (proper) closed normal subgroup and let $\Gamma \leq G$ be a uniform lattice, i.e., a discrete subgroup such that $G/\Gamma$ Is compact.
Suppose ...
user215380
4
votes
1
answer
309
views
On self-duality of non-Archimedean local fields
The question to follow has already been asked by the OP at https://math.stackexchange.com/questions/3454735/on-self-duality-of-non-archimedean-local-fields. Due to a lack of feedback, the OP felt ...
- 353
4
votes
1
answer
495
views
Weil's Haar measure construction from below
Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function.
I would need to know something similar for an ...
user1688
4
votes
0
answers
97
views
Characterization of Vilenkin group
It is shown in [1, Section 1] by C.W. Onneweer that every infinite compact, metrizable, zero-dimensional commutative group is a Vilenkin group. My question is does this implication also hold if we ...
- 85
4
votes
0
answers
284
views
Failure of Schur's lemma for topological group representations
Is there an example of $G$, $\rho$ as below?
$G$ is a locally compact group.
$\rho$ is an irreducible continuous representation of $G$ on a complex Hilbert space $V$. This means that we have a ...
3
votes
2
answers
483
views
When does a LCA group not contain a (closed) infinite cyclic subgroup?
If $G$ is an LCA (locally compact abelian) group, is there any 'nice' sufficient (or preferably necessary and sufficient) criteria for when $G$ does not contain a closed (and hence discrete in the ...
- 3,115
3
votes
1
answer
153
views
Urysohn's lemma for Bochner functions?
Let $G$ be a locally compact Hausdorff group. In the proof of theorem 4.32 of Folland's book "A course in abstract harmonic analysis", the following result is used:
If $U$ is an open ...
- 175
3
votes
1
answer
861
views
Continuous function defined by measurable sets
Is the following slightly generalization of Corollary 20.17 in Hewitt and Ross Book (page 296) correct?
Let $A$ be a subset of a profinite group $G$ ( compact, Hausdorff, totally disconnected ...
- 3,738
3
votes
1
answer
167
views
When are all the convolution roots of an infinitely divisible probability measure infinitely divisible?
Let $G$ be a topological group.
Let us say that a probability measure $\mu$ on $G$ is strongly infinitely divisible (SID) if $\mu$ is infinitely divisible and any probability measure $\nu$ on $G$ ...
- 128k
3
votes
1
answer
347
views
Subgroups of finite non-zero Haar measure of abelian locally compact groups
Is it true that every subgroup of finite non-zero Haar measure of an abelian locally compact group should be open and compact? This is obviously true for the case of discrete abelian groups. Thanks.
- 895
3
votes
0
answers
55
views
Norm under Gelfand map vs norm under left regular representation on $\ell^p$
Let $G$ be a discrete commutative group. Let $p \in [1,\infty)$ and consider the left regular representation $\lambda : \ell^1(G) \to \mathcal{B}(\ell^p(G))$; that is $\lambda(x)y := x*y$,
where
$$
(x*...
- 131
3
votes
0
answers
93
views
About the nilpotency of a subgroup
Let $G$ be a compact group. Let $\mathcal N$ be a family of closed normal subgroups of nilpotency class at most $k$. Assume that $\mathcal N$ is closed under finite intersections and $\bigcap_{N\in\...
- 2,118
3
votes
0
answers
81
views
Quantum analogue of certain property of compact groups
Let $\mathcal{A}$ be the category of $C^*$ algebras. For a group $G$ let $\tilde{G}$ be the space of all conjugacy classes of $G$.
What is a precise description of a maximal ,or in some sense ...
- 356
3
votes
0
answers
105
views
A generalization of the character group
Let $G$ be a group.
We define $$\tilde{G}=\{\phi:G \to \mathbb{T}\mid \phi(gh){\phi(g)}^{-1}{\phi(h)}^{-1}\in Tor(\mathbb{T})\}$$
where $Tor(\mathbb{T})$ is the group of torsion elements of the unit ...
- 356
3
votes
0
answers
284
views
Interior of fundamental domains of lattices in locally compact groups
Let $G$ be a locally compact abelian group, and let $\Lambda$ be a lattice in $G$, i.e. a discrete subgroup such that the quotient group $G/\Lambda$ is compact.
A fundamental domain for $\Lambda$ in $...
- 131
3
votes
0
answers
143
views
Is an Abelian topological group compact if it is complete and Bohr-compact?
A topological group $G$ will be called Bohr-compact if its Bohr topology (i.e., the largest precompact group topology) is compact and Hausdorff.
A topological group $G$ is Bohr-compact if it admits ...
- 41.9k
3
votes
0
answers
317
views
Best constant for maximal function for locally compact groups
Maximal functions for locally compact groups have been studied. In particular papers of K. Phillips and Phillips-Taibleson (here and here) contain nice results in this direction. It might be too much ...
- 1,583
3
votes
0
answers
739
views
Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)
Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as
$$...
- 960
2
votes
1
answer
223
views
Show that $V_G: L^2(G\times G, \mu \times \mu)\to L^2(G\times G, \mu \times \mu)$ defined by $V_G(f)(x,y) = f(xy,y)$ is well-defined
Let $G$ be a locally compact Hausdorff group and let $\mu$ be a right Haar measure on $G$. Then $\mu\times \mu$ (the Radon product of measures) is a right Haar measure on $G \times G$ and we can ...
- 175
2
votes
1
answer
246
views
Examples of non-discrete, cocompact subgroups
I am looking for non-trivial examples of the following:
$G$ is a locally compact group
$H\subset G$ a closed subgroup
Both are unimodular and non-discrete
The quotient space $G/H$ is compact, but $G$ ...
user473423