Pontryagin duality

In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group, the finite abelian groups, and the additive group of the integers, the real numbers, and every finite-dimensional vector space over the reals or a -adic field.
The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological group, consisting of the continuous group homomorphisms from the group to the circle group, with the operation of pointwise multiplication and the topology of uniform convergence on compact sets. The Pontryagin duality theorem establishes Pontryagin duality by stating that any locally compact abelian group is naturally isomorphic with its bidual. The Fourier inversion theorem is a special case of this theorem.
The subject is named after Lev Pontryagin, who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's treatment relied on the groups being second-countable and either compact or discrete. This was improved to cover the general locally compact abelian groups by Egbert van Kampen in 1935 and André Weil in 1940.

Introduction

Pontryagin duality places in a unified context a number of observations about functions on the real line or on finite abelian groups:

Suitably regular complex-valued periodic functions on the real line have Fourier series and these functions can be recovered from their Fourier series;
Suitably regular complex-valued functions on the real line have Fourier transforms that are also functions on the real line and, just as for periodic functions, these functions can be recovered from their Fourier transforms; and
Complex-valued functions on a finite abelian group have discrete Fourier transforms, which are functions on the [|dual group], which is a isomorphic group. Moreover, any function on a finite abelian group can be recovered from its discrete Fourier transform.

The theory, introduced by Lev Pontryagin and combined with the Haar measure introduced by John von Neumann, André Weil and others depends on the theory of the dual group of a locally compact abelian group.
It is analogous to the dual vector space of a vector space: a finite-dimensional vector space and its dual vector space are not naturally isomorphic, but the endomorphism algebra of one is isomorphic to the opposite of the endomorphism algebra of the other: via the transpose. Similarly, a group and its dual group are not in general isomorphic, but their endomorphism rings are opposite to each other:. More categorically, this is not just an isomorphism of endomorphism algebras, but a contravariant equivalence of categories – see .

Definition

A topological group is a locally compact group if the underlying topological space is locally compact and Hausdorff; a topological group is abelian if the underlying group is abelian.
Examples of locally compact abelian groups include finite abelian groups, the integers, the real numbers, the circle group T, and also the p-adic numbers.
For a locally compact abelian group, the Pontryagin dual is the group of continuous group homomorphisms from to the circle group. That is,
The Pontryagin dual is usually endowed with the topology given by uniform convergence on compact sets.

Examples

The Pontryagin dual of a finite cyclic group is isomorphic to itself.
The Pontryagin dual of the group of integers is the circle group, and the Pontryagin dual of the circle group is the group of integers.
The Pontryagin dual of the group of real numbers is itself.
The Pontryagin dual of the group of -adic integers is the Prüfer -group, and the Pontryagin dual of the Prüfer -group is the group of -adic integers.
Pontryagin duality theorem

means that there is a naturally defined map ; more importantly, the map should be functorial in. For the multiplicative character of the group, the canonical isomorphism is defined on as follows:
That is,
In other words, each group element is identified to the evaluation character on the dual. This is strongly analogous to the canonical isomorphism between a finite-dimensional vector space and its double dual,, and it is worth mentioning that any vector space is an abelian group. If is a finite abelian group, then but this isomorphism is not canonical. Making this statement precise requires thinking about dualizing not only on groups, but also on maps between the groups, in order to treat dualization as a functor and prove the identity functor and the dualization functor are not naturally equivalent. Also the duality theorem implies that for any group the dualization functor is an exact functor.

Pontryagin duality and the Fourier transform

Haar measure

One of the most remarkable facts about a locally compact group is that it carries an essentially unique natural measure, the Haar measure, which allows one to consistently measure the "size" of sufficiently regular subsets of. "Sufficiently regular subset" here means a Borel set; that is, an element of the σ-algebra generated by the compact sets. More precisely, a right Haar measure on a locally compact group is a countably additive measure μ defined on the Borel sets of which is right invariant in the sense that for an element of and a Borel subset of and also satisfies some regularity conditions. Except for positive scaling factors, a Haar measure on is unique.
The Haar measure on allows us to define the notion of integral for Borel functions defined on the group. In particular, one may consider various L^p spaces associated to the Haar measure. Specifically,
Note that, since any two Haar measures on are equal up to a scaling factor, this -space is independent of the choice of Haar measure and thus perhaps could be written as. However, the -norm on this space depends on the choice of Haar measure, so if one wants to talk about isometries it is important to keep track of the Haar measure being used.

Fourier transform and Fourier inversion formula for ''L''¹-functions

The dual group of a locally compact abelian group is used as the underlying space for an abstract version of the Fourier transform. If, then the Fourier transform is the function on defined by
where the integral is relative to Haar measure on. This is also denoted. Note the Fourier transform depends on the choice of Haar measure. It is not too difficult to show that the Fourier transform of an function on is a bounded continuous function on which vanishes at infinity.
The inverse Fourier transform of an integrable function on is given by
where the integral is relative to the Haar measure on the dual group. The measure on that appears in the Fourier inversion formula is called the dual measure to and may be denoted.
The various Fourier transforms can be classified in terms of their domain and transform domain as follows :

Transform	Original domain,	Transform domain,	Measure,
Fourier transform
Fourier series
Discrete-time Fourier transform
Discrete Fourier transform

As an example, suppose, so we can think about as by the pairing If is the Lebesgue measure on Euclidean space, we obtain the ordinary Fourier transform on and the dual measure needed for the Fourier inversion formula is. If we want to get a Fourier inversion formula with the same measure on both sides then we need to use
However, if we change the way we identify with its dual group, by using the pairing
then Lebesgue measure on is equal to its own dual measure. This convention minimizes the number of factors of that show up in various places when computing Fourier transforms or inverse Fourier transforms on Euclidean space. Note that the choice of how to identify with its dual group affects the meaning of the term "self-dual function", which is a function on equal to its own Fourier transform: using the classical pairing the function is self-dual. But using the pairing, which keeps the pre-factor as unity, makes self-dual instead. This second definition for the Fourier transform has the advantage that it maps the multiplicative identity to the convolution identity, which is useful as is a convolution algebra. See the next section on [|the group algebra]. In addition, this form is also necessarily isometric on spaces. See below at Plancherel and L² Fourier inversion theorems.

Group algebra

The space of integrable functions on a locally compact abelian group is an algebra, where multiplication is convolution: the convolution of two integrable functions and is defined as
This algebra is referred to as the Group Algebra of. By the Fubini–Tonelli theorem, the convolution is submultiplicative with respect to the norm, making a Banach algebra. The Banach algebra has a multiplicative identity element if and only if is a discrete group, namely the function that is 1 at the identity and zero elsewhere. In general, however, it has an approximate identity which is a net indexed on a directed set such that
The Fourier transform takes convolution to multiplication, i.e. it is a homomorphism of abelian Banach algebras :
In particular, to every group character on corresponds a unique multiplicative linear functional on the group algebra defined by
It is an important property of the group algebra that these exhaust the set of non-trivial multiplicative linear functionals on the group algebra; see section 34 of Loomis. This means the Fourier transform is a special case of the Gelfand transform.