Pontryagin duality

One of the most remarkable facts about a locally compact group ${\displaystyle G}$ 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 ${\displaystyle G}$. "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 ${\displaystyle G}$ is a countably additive measure μ defined on the Borel sets of ${\displaystyle G}$ which is right invariant in the sense that ${\displaystyle \mu (Ax)=\mu (A)}$ for ${\displaystyle x}$ an element of ${\displaystyle G}$ and ${\displaystyle A}$ a Borel subset of ${\displaystyle G}$ and also satisfies some regularity conditions (spelled out in detail in the article on Haar measure). Except for positive scaling factors, a Haar measure on ${\displaystyle G}$ is unique.

The Haar measure on ${\displaystyle G}$ 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 ${\displaystyle \mu }$. Specifically,

$${\displaystyle {\mathcal {L}}_{\mu }^{p}(G)=\left\{(f:G\to \mathbb {C} )\ {\Big |}\ \int _{G}|f(x)|^{p}\ d\mu (x)<\infty \right\}.}$$

Note that, since any two Haar measures on ${\displaystyle G}$ are equal up to a scaling factor, this ${\displaystyle L^{p}}$-space is independent of the choice of Haar measure and thus perhaps could be written as ${\displaystyle L^{p}(G)}$. However, the ${\displaystyle L^{p}}$-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.

The dual group of a locally compact abelian group is used as the underlying space for an abstract version of the Fourier transform. If ${\displaystyle f\in L^{1}(G)}$, then the Fourier transform is the function ${\displaystyle {\widehat {f}}}$ on ${\displaystyle {\widehat {G}}}$ defined by

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

where the integral is relative to Haar measure ${\displaystyle \mu }$ on ${\displaystyle G}$. This is also denoted ${\displaystyle ({\mathcal {F}}f)(\chi )}$. Note the Fourier transform depends on the choice of Haar measure. It is not too difficult to show that the Fourier transform of an ${\displaystyle L^{1}}$ function on ${\displaystyle G}$ is a bounded continuous function on ${\displaystyle {\widehat {G}}}$ which vanishes at infinity.

Fourier Inversion Formula for ${\displaystyle L^{1}}$-Functions — For each Haar measure ${\displaystyle \mu }$ on ${\displaystyle G}$ there is a unique Haar measure ${\displaystyle \nu }$ on ${\displaystyle {\widehat {G}}}$ such that whenever ${\displaystyle f\in L^{1}(G)}$ and ${\displaystyle {\widehat {f}}\in L^{1}\left({\widehat {G}}\right)}$, we have

$${\displaystyle f(x)=\int _{\widehat {G}}{\widehat {f}}(\chi )\chi (x)\ d\nu (\chi )\qquad \mu {\text{-almost everywhere}}}$$

If ${\displaystyle f}$ is continuous then this identity holds for all ${\displaystyle x}$.

The inverse Fourier transform of an integrable function on ${\displaystyle {\widehat {G}}}$ is given by

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

where the integral is relative to the Haar measure ${\displaystyle \nu }$ on the dual group ${\displaystyle {\widehat {G}}}$. The measure ${\displaystyle \nu }$ on ${\displaystyle {\widehat {G}}}$ that appears in the Fourier inversion formula is called the dual measure to ${\displaystyle \mu }$ and may be denoted ${\displaystyle {\widehat {\mu }}}$.

The various Fourier transforms can be classified in terms of their domain and transform domain (the group and dual group) as follows (note that ${\displaystyle \mathbb {T} }$ is Circle group):

Transform	Original domain, ${\displaystyle G}$	Transform domain, ${\displaystyle {\hat {G}}}$	Measure, ${\displaystyle \mu }$
Fourier transform	${\displaystyle \mathbb {R} }$	${\displaystyle \mathbb {R} }$	${\displaystyle {\text{Constant}}\times {\text{Lebesgue measure}}}$
Fourier series	${\displaystyle \mathbb {T} }$	${\displaystyle \mathbb {Z} }$	${\displaystyle {\text{Constant}}\times {\text{Lebesgue measure}}}$
Discrete-time Fourier transform (DTFT)	${\displaystyle \mathbb {Z} }$	${\displaystyle \mathbb {T} }$	${\displaystyle {\text{Constant}}\times {\text{Counting measure}}}$
Discrete Fourier transform (DFT)	${\displaystyle \mathbb {Z} _{n}}$	${\displaystyle \mathbb {Z} _{n}}$	${\displaystyle {\text{Constant}}\times {\text{Counting measure}}}$

As an example, suppose ${\displaystyle G=\mathbb {R} ^{n}}$, so we can think about ${\displaystyle {\widehat {G}}}$ as ${\displaystyle \mathbb {R} ^{n}}$ by the pairing ${\displaystyle (\mathbf {v} ,\mathbf {w} )\mapsto e^{i\mathbf {v} \cdot \mathbf {w} }.}$ If ${\displaystyle \mu }$ is the Lebesgue measure on Euclidean space, we obtain the ordinary Fourier transform on ${\displaystyle \mathbb {R} ^{n}}$ and the dual measure needed for the Fourier inversion formula is ${\displaystyle {\widehat {\mu }}=(2\pi )^{-n}\mu }$. If we want to get a Fourier inversion formula with the same measure on both sides (that is, since we can think about ${\displaystyle \mathbb {R} ^{n}}$ as its own dual space we can ask for ${\displaystyle {\widehat {\mu }}}$ to equal ${\displaystyle \mu }$) then we need to use

$${\displaystyle {\begin{aligned}\mu &=(2\pi )^{-{\frac {n}{2}}}\times {\text{Lebesgue measure}}\\{\widehat {\mu }}&=(2\pi )^{-{\frac {n}{2}}}\times {\text{Lebesgue measure}}\end{aligned}}}$$

However, if we change the way we identify ${\displaystyle \mathbb {R} ^{n}}$ with its dual group, by using the pairing

$${\displaystyle (\mathbf {v} ,\mathbf {w} )\mapsto e^{2\pi i\mathbf {v} \cdot \mathbf {w} },}$$

then Lebesgue measure on ${\displaystyle \mathbb {R} ^{n}}$ is equal to its own dual measure. This convention minimizes the number of factors of ${\displaystyle 2\pi }$ that show up in various places when computing Fourier transforms or inverse Fourier transforms on Euclidean space. (In effect it limits the ${\displaystyle 2\pi }$ only to the exponent rather than as a pre-factor outside the integral sign.) Note that the choice of how to identify ${\displaystyle \mathbb {R} ^{n}}$ with its dual group affects the meaning of the term "self-dual function", which is a function on ${\displaystyle \mathbb {R} ^{n}}$ equal to its own Fourier transform: using the classical pairing ${\displaystyle (\mathbf {v} ,\mathbf {w} )\mapsto e^{i\mathbf {v} \cdot \mathbf {w} }}$ the function ${\displaystyle e^{-{\frac {1}{2}}x^{2}}}$ is self-dual. But using the pairing, which keeps the pre-factor as unity, ${\displaystyle (\mathbf {v} ,\mathbf {w} )\mapsto e^{2\pi i\mathbf {v} \cdot \mathbf {w} }}$ makes ${\displaystyle e^{-\pi x^{2}}}$ 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 ${\displaystyle L^{1}}$ is a convolution algebra. See the next section on the group algebra. In addition, this form is also necessarily isometric on ${\displaystyle L^{2}}$ spaces. See below at Plancherel and L² Fourier inversion theorems.

The space of integrable functions on a locally compact abelian group ${\displaystyle G}$ is an algebra, where multiplication is convolution: the convolution of two integrable functions ${\displaystyle f}$ and ${\displaystyle g}$ is defined as

$${\displaystyle (f*g)(x)=\int _{G}f(x-y)g(y)\ d\mu (y).}$$

This algebra is referred to as the Group Algebra of ${\displaystyle G}$. By the Fubini–Tonelli theorem, the convolution is submultiplicative with respect to the ${\displaystyle L^{1}}$ norm, making ${\displaystyle L^{1}(G)}$ a Banach algebra. The Banach algebra ${\displaystyle L^{1}(G)}$ has a multiplicative identity element if and only if ${\displaystyle G}$ 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 (or generalized sequence) ${\displaystyle \{e_{i}\}_{i\in I}}$ indexed on a directed set ${\displaystyle I}$ such that ${\displaystyle f*e_{i}\to f.}$

The Fourier transform takes convolution to multiplication, i.e. it is a homomorphism of abelian Banach algebras ${\displaystyle L^{1}(G)\to C_{0}\left({\widehat {G}}\right)}$ (of norm ≤ 1):

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

In particular, to every group character on ${\displaystyle G}$ corresponds a unique multiplicative linear functional on the group algebra defined by

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

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 (Loomis 1953). This means the Fourier transform is a special case of the Gelfand transform.

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 — Choose a Haar measure ${\displaystyle \mu }$ on ${\displaystyle G}$ and let ${\displaystyle \nu }$ be the dual measure on ${\displaystyle {\widehat {G}}}$ as defined above. If ${\displaystyle f:G\to \mathbb {C} }$ is continuous with compact support then ${\displaystyle {\widehat {f}}\in L^{2}\left({\widehat {G}}\right)}$ and

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

In particular, the Fourier transform is an ${\displaystyle L^{2}}$ isometry from the complex-valued continuous functions of compact support on ${\displaystyle G}$ to the ${\displaystyle L^{2}}$-functions on ${\displaystyle {\widehat {G}}}$ (using the ${\displaystyle L^{2}}$-norm with respect to ${\displaystyle \mu }$ for functions on ${\displaystyle G}$ and the ${\displaystyle L^{2}}$-norm with respect to ${\displaystyle \nu }$ for functions on ${\displaystyle {\widehat {G}}}$).

Since the complex-valued continuous functions of compact support on ${\displaystyle G}$ are ${\displaystyle L^{2}}$-dense, there is a unique extension of the Fourier transform from that space to a unitary operator

$${\displaystyle {\mathcal {F}}:L_{\mu }^{2}(G)\to L_{\nu }^{2}\left({\widehat {G}}\right).}$$

and we have the formula

$${\displaystyle \forall f\in L^{2}(G):\quad \int _{G}|f(x)|^{2}\ d\mu (x)=\int _{\widehat {G}}\left|{\widehat {f}}(\chi )\right|^{2}\ d\nu (\chi ).}$$

Note that for non-compact locally compact groups ${\displaystyle G}$ the space ${\displaystyle L^{1}(G)}$ does not contain ${\displaystyle L^{2}(G)}$, so the Fourier transform of general ${\displaystyle L^{2}}$-functions on ${\displaystyle G}$ is "not" given by any kind of integration formula (or really any explicit formula). To define the ${\displaystyle L^{2}}$ Fourier transform one has to resort to some technical trick such as starting on a dense subspace like the continuous functions with compact support and then extending the isometry by continuity to the whole space. This unitary extension of the Fourier transform is what we mean by 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 ${\displaystyle L^{2}}$ Fourier transform. This is the content of the ${\displaystyle L^{2}}$ Fourier inversion formula which follows.

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

$${\displaystyle L_{\nu }^{2}\left({\widehat {G}}\right)\to L_{\mu }^{2}(G)}$$

where ${\displaystyle \nu }$ is the dual measure to ${\displaystyle \mu }$.

In the case ${\displaystyle G=\mathbb {T} }$ the dual group ${\displaystyle {\widehat {G}}}$ is naturally isomorphic to the group of integers ${\displaystyle \mathbb {Z} }$ and the Fourier transform specializes to the computation of coefficients of Fourier series of periodic functions.

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

When ${\displaystyle G}$ is a Hausdorff abelian topological group, the group ${\displaystyle {\widehat {G}}}$ with the compact-open topology is a Hausdorff abelian topological group and the natural mapping from ${\displaystyle G}$ to its double-dual ${\displaystyle {\widehat {\widehat {G}}}}$ makes sense. If this mapping is an isomorphism, it is said that ${\displaystyle G}$ satisfies Pontryagin duality (or that ${\displaystyle G}$ is a reflexive group,[4] or a reflective group[5]). This has been extended in a number of directions beyond the case that ${\displaystyle G}$ is locally compact.[6]

In particular, Samuel Kaplan[7][8] showed in 1948 and 1950 that arbitrary products and countable inverse limits of locally compact (Hausdorff) abelian groups satisfy Pontryagin duality. Note that an infinite product of locally compact non-compact spaces is not locally compact.

Later, in 1975, Rangachari Venkataraman[9] showed, among other facts, that every open subgroup of an abelian topological group which satisfies Pontryagin duality itself satisfies Pontryagin duality.

More recently, Sergio Ardanza-Trevijano and María Jesús Chasco[10] have extended the results of Kaplan mentioned above. They showed that direct and inverse limits of sequences of abelian groups satisfying Pontryagin duality also satisfy Pontryagin duality if the groups are metrizable or ${\displaystyle k_{\omega }}$-spaces but not necessarily locally compact, provided some extra conditions are satisfied by the sequences.

However, there is a fundamental aspect that changes if we want to consider Pontryagin duality beyond the locally compact case. Elena Martín-Peinador[11] proved in 1995 that if ${\displaystyle G}$ is a Hausdorff abelian topological group that satisfies Pontryagin duality, and the natural evaluation pairing

$${\displaystyle {\begin{cases}G\times {\widehat {G}}\to \mathbb {T} \\(x,\chi )\mapsto \chi (x)\end{cases}}}$$

is (jointly) continuous,[lower-alpha 1] then ${\displaystyle G}$ is locally compact. As a corollary, all non-locally compact examples of Pontryagin duality are groups where the pairing ${\displaystyle G\times {\widehat {G}}\to \mathbb {T} }$ is not (jointly) continuous.

Another way to generalize Pontryagin duality to wider classes of commutative topological groups is to endow the dual group ${\displaystyle {\widehat {G}}}$ with a bit different topology, namely the topology of uniform convergence on totally bounded sets. The groups satisfying the identity ${\displaystyle G\cong {\widehat {\widehat {G}}}}$ under this assumption[lower-alpha 2] are called stereotype groups.[5] This class is also very wide (and it contains locally compact abelian groups), but it is narrower than the class of reflective groups.[5]

In 1952 Marianne F. Smith[12] noticed that Banach spaces and reflexive spaces, being considered as topological groups (with the additive group operation), satisfy Pontryagin duality. Later B. S. Brudovskiĭ,[13] William C. Waterhouse[14] and K. Brauner[15] showed that this result can be extended to the class of all quasi-complete barreled spaces (in particular, to all Fréchet spaces). In the 1990s Sergei Akbarov[16] gave a description of the class of the topological vector spaces that satisfy a stronger property than the classical Pontryagin reflexivity, namely, the identity

$${\displaystyle (X^{\star })^{\star }\cong X}$$

where ${\displaystyle X^{\star }}$ means the space of all linear continuous functionals ${\displaystyle f\colon X\to \mathbb {C} }$ endowed with the topology of uniform convergence on totally bounded sets in ${\displaystyle X}$ (and ${\displaystyle (X^{\star })^{\star }}$ means the dual to ${\displaystyle X^{\star }}$ in the same sense). The spaces of this class are called stereotype spaces, and the corresponding theory found a series of applications in Functional analysis and Geometry, including the generalization of Pontryagin duality for non-commutative topological groups.

For non-commutative locally compact groups ${\displaystyle G}$ the classical Pontryagin construction stops working for various reasons, in particular, because the characters don't always separate the points of ${\displaystyle G}$, and the irreducible representations of ${\displaystyle G}$ are not always one-dimensional. At the same time it is not clear how to introduce multiplication on the set of irreducible unitary representations of ${\displaystyle G}$, and it is even not clear whether this set is a good choice for the role of the dual object for ${\displaystyle G}$. So the problem of constructing duality in this situation requires complete rethinking.

Theories built to date are divided into two main groups: the theories where the dual object has the same nature as the source one (like in the Pontryagin duality itself), and the theories where the source object and its dual differ from each other so radically that it is impossible to count them as objects of one class.

The second type theories were historically the first: soon after Pontryagin's work Tadao Tannaka (1938) and Mark Krein (1949) constructed a duality theory for arbitrary compact groups known now as the Tannaka–Krein duality.[17][18] In this theory the dual object for a group ${\displaystyle G}$ is not a group but a category of its representations ${\displaystyle \Pi (G)}$.

Duality for finite groups.

The theories of first type appeared later and the key example for them was the duality theory for finite groups.[19][20] In this theory the category of finite groups is embedded by the operation ${\displaystyle G\mapsto \mathbb {C} _{G}}$ of taking group algebra ${\displaystyle \mathbb {C} _{G}}$ (over ${\displaystyle \mathbb {C} }$) into the category of finite dimensional Hopf algebras, so that the Pontryagin duality functor ${\displaystyle G\mapsto {\widehat {G}}}$ turns into the operation ${\displaystyle H\mapsto H^{*}}$ of taking the dual vector space (which is a duality functor in the category of finite dimensional Hopf algebras).[20]

In 1973 Leonid I. Vainerman, George I. Kac, Michel Enock, and Jean-Marie Schwartz built a general theory of this type for all locally compact groups.[21] From the 1980s the research in this area was resumed after the discovery of quantum groups, to which the constructed theories began to be actively transferred.[22] These theories are formulated in the language of C*-algebras, or Von Neumann algebras, and one of its variants is the recent theory of locally compact quantum groups.[23][22]

One of the drawbacks of these general theories, however, is that in them the objects generalizing the concept of a group are not Hopf algebras in the usual algebraic sense.[20] This deficiency can be corrected (for some classes of groups) within the framework of duality theories constructed on the basis of the notion of envelope of topological algebra.[24]