Pontryagin duality

In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. It 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 (non-canonically) isomorphic group. Moreover any function on a finite group can be recovered from its discrete Fourier transform.

The theory, introduced by Lev Pontryagin and combined with 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.

Haar measure

A topological group is locally compact if and only if the identity e of the group has a compact neighborhood. This means that there is some open set V containing e whose closure is relatively compact in the topology of G. One of the most remarkable facts about a locally compact group G 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 G. "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 G is a countably additive measure μ defined on the Borel sets of G which is right invariant in the sense that μ(A x) = μ(A) for x an element of and A a Borel subset of G and also satisfies some regularity conditions (spelled out in detail in the article Haar measure). Except for positive scale factors, Haar measures are unique.

The Haar measure allows us to define the notion of integral for (complex-valued) Borel functions defined on the group. In particular, one may consider various L^p spaces associated to the Haar measure. Specifically,

$L_{\mu }^{p}(G)=\left\{f:G\rightarrow \mathbb {C} :\int _{G}|f(x)|^{p}\,d\mu (x)<\infty \right\}$

Examples of locally compact abelian groups are:

• Rⁿ, for n a positive integer, with vector addition as group operation.

• The positive real numbers with multiplication as operation. This group is clearly seen to be isomorphic to R. In fact, the exponential mapping implements that isomorphism.

• Any finite abelian group, with the discrete topology. By the structure theorem for finite abelian groups, all such groups are products of cyclic groups.

• The integers Z under addition, again with the discrete topology.

• The circle group, denoted T. This is the group of complex numbers of modulus 1. T is isomorphic as a topological group to the quotient group R/Z .

• The field Q_p of p-adic numbers under addition, with the usual p-adic topology.

The dual group

If G is a locally compact abelian group, a character of G is a continuous group homomorphism from G with values in the circle group T. It can be shown that the set of all characters on G is itself a locally compact abelian group, called the dual group of G. The group operation on the dual group is given by pointwise multiplication of characters, the inverse of a character is its complex conjugate and the topology on the space of characters is that of uniform convergence on compact sets (i.e., the compact-open topology). This topology in general is not metrizable. However, if the group G is a separable locally compact abelian group, then the dual group is metrizable. The dual group of an abelian group G is denoted G^.

Theorem The dual of G^ is canonically isomorphic to G, that is (G^)^ = G in a canonical way.

Canonical means that there is naturally defined map from G into (G^)^; more importantly, the map should be functorial. The precise formulation of this idea involves the concept of natural transformation. This fact is important; for instance, any finite abelian group is isomorphic to its dual, but the isomorphism is not canonical. The canonical isomorphism is defined as follows:

$x\mapsto \{\chi \mapsto \chi (x)\}$

In other words, each group element x is identified to the evaluation character on the dual.

Examples

A character on the infinite cyclic group of integers Z under addition is determined by its value at the generator 1. Thus for any character χ on Z, χ(n)=χ(1)ⁿ. Moreover, this formula defines a character for any choice of χ(1) in T. Thus it follows easily that algebraically the dual of Z is isomorphic to the circle group T. The topology of uniform convergence on compact sets is in this case the topology of pointwise convergence. It is also easily shown that this is the topology of the circle group inherited from the complex numbers.

Hence the dual group of Z is canonically isomorphic with T.

Conversely, a character on T is of the form z → zⁿ for n an integer. Since T is compact, the topology on the dual group is that of uniform convergence, which turns out to be the discrete topology. As a consequence of this, the dual of T is canonically isomorphic with Z.

The group of real numbers R, is isomorphic to its own dual; the characters on R are of the form r → e ^{i θ r}. With these dualities, the version of the Fourier transform to be introduced next coincides with the classical Fourier transform on R.

Fourier transform

The dual group of a locally compact abelian group is introduced as the underlying space for an abstract version of the Fourier transform. If a function is in L¹(G), then the Fourier transform is the function on G^ such that

${\widehat {f}}(\chi )=\int _{G}f(x){\overline {\chi (x)}}\;d\mu (x)$

where the integral is relative to Haar measure μ on G. It is not too difficult to show that the Fourier transform of an L¹ function on G is a bounded continuous function on G^ which vanishes at infinity. Similarly, the inverse Fourier transform of an integrable function on G^ is given by

${\check {g}}(x)=\int _{\widehat {G}}g(\chi )\chi (x)\;d\nu (\chi )$

where the integral is relative to the Haar measure ν on the dual group G^.

The group algebra

The space of integrable functions on a locally compact abelian group G is an algebra, where multiplication is convolution: if f, g are integrable functions then the convolution of f and g is defined as

$[f\star g](x)=\int _{G}f(x-y)g(y)\,d\mu (y).$

Theorem The Banach space L¹(G) is an associative and commutative algebra under convolution.

This algebra is referred to as the Group Algebra of G. By completeness of L¹(G), it is a Banach algebra. The Banach algebra L¹(G) does not have a multiplicative identity element unless G is a discrete group. In general, however, it has an approximate identity which is a net (or generalized sequence) indexed on a directed set I, {e_i}_i with the property that

$f\star e_{i}\rightarrow f.$

The Fourier transform takes convolution to multiplication, that is:

${\mathcal {F}}(f\star g)(\chi )={\mathcal {F}}(f)(\chi )\cdot {\mathcal {F}}(g)(\chi ).$

In particular, to every group character on G corresponds a unique multiplicative linear functional on the group algebra defined by

$f\mapsto {\widehat {f}}(\chi ).$

It is an important property of the group algebra that these exhaust the set of non-trivial (that is, not identically zero) multiplicative linear functionals on the group algebra. See section 34 of the Loomis reference.

Plancherel and Fourier inversion theorems

As we have stated, the dual group of a locally compact abelian group is a locally compact abelian group in its own right and thus has a Haar measure, or more precisely a whole family of scale-related Haar measures.

Theorem. There is a scaling of Haar measure on the dual group so that the Fourier transform restricted to continuous functions of compact support on G, is an isometric linear map. It has a unique extension to a unitary operator

${\mathcal {F}}:L_{\mu }^{2}(G)\rightarrow L_{\nu }^{2}({\widehat {G}})$

where $\nu$ is the Haar measure on the dual group.

Note that for non-compact locally compact groups G the space L¹(G) does not contain L²(G), so one has to resort to some technical trick such as restricting to a dense subspace.

Following the Loomis reference below, we say that Haar measures on G and G^ are associated if and only if the Fourier inversion formula holds. The unitary character of the Fourier transform implies:

$\int _{G}|f(x)|^{2}\ d\mu (x)=\int _{\widehat {G}}|{\widehat {f}}(\chi )|^{2}\ d\nu (\chi )$

for every continuous complex-valued function of compact support on G.

It is the unitary extension of the Fourier transform which we consider to be the Fourier transform on the space of square integrable functions. The dual group also has an inverse Fourier transform in its own right; it can be characterized as the inverse (or adjoint, since it is unitary) of the Fourier transform. This is the content of the Fourier inversion formula which follows.

Theorem. The adjoint of the Fourier transform restricted to continuous functions of compact support is the inverse Fourier transform

$L_{\nu }^{2}({\widehat {G}})\rightarrow L_{\mu }^{2}(G)$

where the measures on G and G^ are associated.

In the case of G = Rⁿ, we have G′ = Rⁿ and we recover the ordinary Fourier transform on the Rⁿ by taking

$\mu =(2\pi )^{-n/2}\times {\mbox{Lebesgue measure}}$

$\nu =(2\pi )^{-n/2}\times {\mbox{Lebesgue measure}}$

In the case G = T, the dual group G′ is naturally isomorphic to the group of integers Z and the above operator F specializes to the computation of coefficients of Fourier series of periodic functions.

If G is a finite group, we recover the discrete Fourier transform. Note that this case is very easy to prove directly.

Bohr compactification and almost-periodicity

One important application of Pontryagin duality is the following characterization of compact abelian topological groups:

Theorem. A locally compact abelian group G is compact if and only if the dual group G^{^} is discrete. Conversely, G is discrete if and only if G^{^} is compact.

The Bohr compactification is defined for any topological group G, regardless of whether G is locally compact or abelian. One use made of Pontryagin duality between compact abelian groups and discrete abelian groups is to characterize the Bohr compactification of an arbitrary abelian locally compact topological group. The Bohr compactification B(G) of G is H^{^}, where H has the group structure G^{^}, but given the discrete topology. Since the inclusion map

$\iota :H\rightarrow {\widehat {G}}$

is continuous and a homomorphism, the dual morphism

$G\sim {\widehat {\widehat {G}}}{\rightarrow }{\widehat {H}}$

is a morphism into a compact group which is easily shown to satisfy the requisite universal property.

See also almost periodic function.

Categorical considerations

It is useful to regard the dual group functorially. In what follows, LCA is the category of locally compact abelian groups and continuous group homomorphisms. The dual group construction of G^{^} is a contravariant functor LCA → LCA. In particular, the iterated functor G → (G^{^})^{^} is covariant.

Theorem. The dual group is a category isomorphism from LCA to LCA^op.

Theorem. The iterated dual functor is naturally isomorphic to the identity functor on LCA.

This isomorphism is comparable to the double dual of finite-dimensional vector spaces (a special case, for real and complex vector spaces).

The duality interchanges the subcategories of discrete groups and compact groups. If R is a ring and G is a left R-module, the dual group G^ will become a right R-module; in this way we can also see that discrete left R-modules will be Pontryagin dual to compact right R-modules. The ring End(G) of endomorphisms in LCA is changed by duality into its opposite ring (change the multiplication to the other order). For example if G is an infinite cyclic discrete group, G^ is a circle group: the former has End(G) = Z so this is true also of the latter.

Non-commutative theory

Such a theory cannot exist in the same form for non-commutative groups G, since in that case the appropriate dual object G^{^} of isomorphism classes of representations cannot only contain one-dimensional representations, and will fail to be a group. The generalisation that has been found useful in category theory is called Tannaka-Krein duality; but this diverges from the connection with harmonic analysis, which needs to tackle the question of the Plancherel measure on G^{^}.

There are analogues of duality theory for noncommutative groups, some of which are formulated in the language of C*- algebras.

History

The foundations for the theory of locally compact abelian groups and their duality was laid down by Lev Semenovich Pontryagin in 1934. His treatment relied on the group being second-countable and either compact or discrete. This was improved to cover the general locally compact abelian groups by E.R. van Kampen in 1935 and André Weil in 1953.

References

The following books (available in most university libraries) have chapters on locally compact abelian groups, duality and Fourier transform. The Dixmier reference (also available in English translation) has material on non-commutative harmonic analysis.

• Jacques Dixmier, Les C*-algèbres et leurs Représentations, Gauthier-Villars,1969.
• Lynn H. Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand Co, 1953
• Walter Rudin, Fourier Analysis on Groups, 1962
• Hans Reiter, Classical Harmonic Analysis and Locally Compact Groups, 1968 (2nd ed produced by Jan D. Stegeman, 2000).
• Hewitt and Ross, Abstract Harmonic Analysis, vol 1, 1963.