Questions tagged [pontrjagin-duality]
The pontrjagin-duality tag has no usage guidance.
33 questions
6
votes
0
answers
74
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
1
vote
1
answer
101
views
Cohomology of "symplectically self-dual" chain complex
Let $G,H$ be abelian groups (denoted additively) with their Pontryagin duals denoted as $G^*$ and $H^*$. (The cases I'm interested in are products of $\mathbb Z_n$, $\mathbb Z$, $\mathbb R$, and $\...
- 3,001
8
votes
1
answer
588
views
Is there an explicit construction of the Bohr Compactification of the Integers?
Is it possible to explicitly describe the Bohr compactification of $\mathbb Z$? This is equivalent to describing all the group homomorphisms $\mathbb R/\mathbb Z \to \mathbb R/\mathbb Z$ including ...
- 1,955
5
votes
1
answer
249
views
Double hom with $\mathbb{CP}^\infty$
Pontrjagin duality gives a double dual theorem for "hom with $S^1$", and $S^1$ is $\textbf{B}\mathbb{Z}$ up to homotopy.
$\textbf{B} \textbf{B}\mathbb{Z}$, modeled by $\mathbb{C}\mathbb{P}^{\...
user30211
1
vote
0
answers
88
views
When is A ⊗ ℚ self-Pontrjagin dual for a compact-Hausdorff topological ring A?
The topological ring of finite adeles $\mathbb A \cong \hat{\mathbb Z} \otimes \mathbb Q$ is self-Pontrjagin dual with self-dual Schwartz–Bruhat functional $\mathbb 1_{\hat{\mathbb Z}}$. This ...
user30211
4
votes
0
answers
220
views
Bochner theorem for (non-abelian) discrete groups
I am interested in Pontryagin duality-like theories for discrete groups, more particularly, whether an analogue to Bochner's theorem for abelian groups exists in the discrete non-finite and non-...
- 261
1
vote
0
answers
155
views
Can a nontrivial abelian group have trivial Pontryagin dual? [closed]
Let $A$ be an abelian group, and suppose $$\mathrm{Hom}(A,\mathbb{Q}/\mathbb{Z})=0.$$
Does it follow that $A=0$?
This is true for $A$ finitely-generated, any subgroup of any product of copies of $\...
- 15.4k
6
votes
1
answer
321
views
Topologies that turn the real numbers into a compact Hausdorff topological group
If I'm not mistaken, the question I put on the title used to be on this site, but I'm not being able to find it at all. I'm therefore reposting it so that someone can either give me the old link or ...
- 162
8
votes
1
answer
392
views
Pontryagin dual of a group-cohomology class
Let $A, B, C$ be finite Abelian groups fitting in a short exact sequence
$$
1 \rightarrow A\overset{\iota}{\rightarrow} B\overset{\pi}{\rightarrow} C\rightarrow 1
$$
This determines a class $[\...
- 813
2
votes
0
answers
103
views
Morphism of discrete quantum groups
In the paper Kazhdan's Property T for Discrete Quantum Groups
, we read the following fragment:
First, note that I think there is a typo and that codomain and domain of the dual maps have to be ...
- 175
1
vote
1
answer
198
views
A map in group cohomology from $H^n(G,G^{\vee})$ to $H^{n+1}(G,U(1))$
Let $G$ be a finite abelian group and denote by $G^{\vee}=\mathrm{Hom}(G,U(1))$ its Pontryagin dual. For any positive integer $n$ one can define a homomorphism of abelian groups
$$
f:H^{n}(G,G^{\vee})\...
- 813
3
votes
1
answer
160
views
Linking form for homology with general coefficients
For integral homology groups there is the notion of linking form (http://www.map.mpim-bonn.mpg.de/Linking_form)
$$
Tor(H_{l}(X,\mathbb{Z}))\times Tor(H_{n-l-1}(X,\mathbb{Z}))\rightarrow \mathbb{Q}/\...
- 813
3
votes
1
answer
368
views
Local Tate duality for F-vector space
A version of local Tate duality stated: Let $K$ be a finite extension of $\mathbb Q_p$, $A$ be a finite $G_K=Gal(\overline K/K)$ module. Then for $0\le i\le 2$, the cup product induces a perfect ...
- 183
7
votes
3
answers
1k
views
Condensed Pontryagin duality
Has Pontryagin duality been extended to condensed abelian groups? The obvious approach being to define $\hat M$ as the internal hom to the circle group. Is it true that $\hat{\hat M}=M$ with this ...
user473423
1
vote
1
answer
257
views
Pontrjagin dual of modules [closed]
I am not sure whether this question is appropriate to appear here. If not, I apologize for that.
Given an $R$-module $M$, we define its Pontrjagin dual as $M^{\ast}=Hom_{\mathbb{Z}}(M, \mathbb{Q/Z})$. ...
- 11
2
votes
2
answers
250
views
Pontryagin-reflexivity of spaces of continuous functions
It is well-known that a Banach space $V$ is always Pontryagin-reflexive, i.e. the natural map $V\to \text{Hom}_\mathbb{R}(\text{Hom}_\mathbb{R}(V, \mathbb{R}), \mathbb{R})$, where $\text{Hom}_\mathbb{...
user103549
1
vote
0
answers
67
views
Pontryagin's principle with Lebesgue-integrable control
Does there exist a (weak) version of Pontryagin's minimum principle in which the control is allowed to be just Lebesgue integrable? I am mostly familiar with the 1975 text of Fleming & Rishel, ...
- 1,253
4
votes
2
answers
182
views
Measure algebra on the Bohr compactification vs the bidual algebras
The following question probably reduces to some standard abstract harmonic analysis Twister play, but I'd still welcome some comments on it.
Let $G$ be a locally compact Abelian group and let $bG$ ...
- 11.3k
7
votes
0
answers
223
views
Duality of Hopf algebras and duality of spectra
Let $S$ be the sphere spectrum, and for $X$ a topological space, let $S(X)$ be the mapping spectrum from the free loop spectrum on $X$ to the sphere spectrum. This is an $E_\infty$ ring spectrum (also ...
- 7,952
4
votes
0
answers
545
views
Is Serre duality related to Pontryagin duality?
I am wondering if there is some relationship between Serre duality and Pontryagin duality for compact complex manifolds. In this case Serre duality reduces to the commutativity of Hodge-star operator ...
- 6,249
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
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
2
votes
0
answers
284
views
Conceptual explanation for Poisson summation formula
The Poisson summation formula says that for a Schwartz function $f : \mathbf R^d \to \mathbf R$ and its Fourier transform $\widehat f$, we have
$$\sum_{n \in \mathbf Z^d} f(x) = \sum_{n \in \mathbf Z^...
- 2,040
2
votes
1
answer
296
views
Creating Duals in A Category
Before stating my question I would like to provide afew motivating examples:
Examples:
In the category of Finitely-generated-projective $R$-modules, we have that:
$M^{\vee}:=Hom_R(M,R)$ satisfies: $...
- 5,407
2
votes
1
answer
299
views
Bohr compactification and "discretization"
Let $G$ is a compact group. We can form the Pontriagin dual $\widehat{G}$ of $G$: it is then discrete space. One can consider the Bohr compactification $b\widehat{G}$ of $\widehat{G}$ which is compact ...
- 9,330
2
votes
1
answer
374
views
What is the dual of a pre-injective map?
In [M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math.
Soc. (JEMS) 1 (1999), 109–197], Gromov introduces the notion of pre-injective map. Recasting this notion in the setting of ...
- 2,452
5
votes
1
answer
2k
views
Proof that the Pontryagin dual of a topological group is a topological group
I'm looking for a proof that the Pontryagin dual $G^*$ of a topological group $G$ is a topological group.
It's very easy to prove that $G^*$ is a group, my troubles are in proving that the map $G^* \...
- 285
6
votes
4
answers
5k
views
Quick computation of the Pontryagin dual group of torus
I'm looking for a quick way to compute the Pontryagin dual group of the n-dimensional torus $\mathbb{T}^n$ (with $\mathbb{T} := \mathbb{R} / \mathbb{Z}$). The only way I know is from "Dikran Dikranjan ...
- 285
19
votes
6
answers
2k
views
Discrete-compact duality for nonabelian groups
A standard property of Pontrjagin duality is that a locally compact Hausdorff abelian group is discrete iff its dual is compact (and vice versa). In what senses, if any, is this still true for ...
- 118k
1
vote
1
answer
1k
views
Fourier Transform of measure on Banach Space (a question about Pontryagin Duality)
The following definition is given as the Fourier transform of a Borel probability measure $\mu$ on $E$, a Banach Space (Real):
$\hat{\mu}: E^*\rightarrow \mathbb{C}$ defined by
$\hat{\mu}(x^*):=\...
- 125
12
votes
6
answers
11k
views
Two reference requests: Pinsker's inequality and Pontryagin duality
Sorry for such a newbie post and for asking two unrelated references in one shot.
First, I am interested in any proof of Pinsker's inequality.
Second, I wonder what is the best place to read about ...
- 1,791
15
votes
1
answer
13k
views
Fourier transforms of compactly supported functions
One manifestation of the uncertainty principle is the fact that a compactly supported function $f$ cannot have a Fourier transform which vanishes on an open set. As stated, this phenomenon applies ...
- 2,243
5
votes
2
answers
1k
views
Injective modules and Pontrjagin duals
Forgive me for this naive question.
We consider the following lemma and its proof in Lang's algebra, Third Ed., published 1999, Chap. 20, section 4, page 784.
Every module is a submodule of an ...
- 7,442