# Questions tagged [pontrjagin-duality]

The pontrjagin-duality tag has no usage guidance.

26
questions

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### 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 ...

7
votes

1
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### 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 $[\...

2
votes

0
answers

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### 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 ...

1
vote

1
answer

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### 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})\...

3
votes

1
answer

93
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### 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}/\...

3
votes

1
answer

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### 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 ...

7
votes

3
answers

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### 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 ...

1
vote

1
answer

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### 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 $M^{\ast}=Hom_{\mathbb{Z}}(M, \mathbb{Q/Z})$. ...

2
votes

2
answers

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### 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{...

1
vote

0
answers

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### 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, ...

4
votes

2
answers

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### 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 $bG$ ...

7
votes

0
answers

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### 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 ...

4
votes

0
answers

498
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### 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
votes

1
answer

374
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 $G^{\wedge}$ ...

16
votes

1
answer

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### 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 $\Gamma_2$ ...

2
votes

0
answers

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### 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
votes

1
answer

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### 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: $...

2
votes

1
answer

266
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 ...

2
votes

1
answer

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### 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
votes

1
answer

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### 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^* \...

6
votes

4
answers

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### 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 ...

18
votes

6
answers

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### 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 ...

1
vote

1
answer

1k
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### 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^*):=\...

12
votes

6
answers

9k
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### 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 ...

14
votes

1
answer

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### 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 ...

5
votes

2
answers

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### 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 ...