In mathematics, Pontryagin duality is a
duality between
locally compact abelian group
In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped with the ...
s that allows generalizing
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
to all such groups, which include the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
(the multiplicative group of complex numbers of modulus one), the
finite abelian groups (with the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
), and the
additive group
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.
This terminology is widely used with structure ...
of the integers (also with the discrete topology), 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 formed by the continuous
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
...
s from the group to the circle group with the operation of pointwise multiplication and the topology of
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
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 dual of its dual). The
Fourier inversion theorem In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information a ...
is a special case of this theorem.
The subject is named after
Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely due ...
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
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
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
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
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 function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
s on the real line have
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
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 transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comp ...
s, which are functions on the
dual group, which is a (non-canonically) isomorphic group. Moreover, any function on a finite abelian group can be recovered from its discrete Fourier transform.
The theory, introduced by
Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely due ...
and combined with the
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, though ...
introduced by
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
,
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
and others depends on the theory of the
dual group of a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ...
abelian group.
It is analogous to the
dual vector space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of a vector space: a finite-dimensional vector space ''V'' and its dual vector space ''V*'' are not naturally isomorphic, but the
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
algebra (matrix 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
categorical considerations.
Definition
A
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
is a
locally compact group
In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...
if the underlying topological space is
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ...
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 (both for the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
, which is also induced by the usual metric), the real numbers, the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
''T'' (both with their usual metric topology), and also the
''p''-adic numbers (with their usual ''p''-adic topology).
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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
given by
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
on
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
s (that is, the topology induced by the
compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory an ...
on the space of all continuous functions from
to
).
For example,
The Pontryagin duality theorem
Canonical
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical exampl ...
means that there is a naturally defined map
; more importantly, the map should be
functorial
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
in
. 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
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to dist ...
and its
double dual
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
,
, and it is worth mentioning that any vector space
is an
Abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
. If
is a finite abelian group, then
but this isomorphism is not canonical. Making this statement precise (in general) requires thinking about dualizing not only on groups, but also on maps between the groups, in order to treat dualization as a
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
and prove the identity functor and the dualization functor are not naturally equivalent. Also the duality theorem implies that for any group (not necessarily finite) the dualization functor is an
exact functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much o ...
.
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 In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, though ...
, which allows one to consistently measure the "size" of sufficiently regular subsets of
. "Sufficiently regular subset" here means a
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
; that is, an element of the
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
generated by the
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
s. 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 (spelled out in detail in the article on
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, though ...
). Except for positive scaling factors, a Haar measure on
is unique.
The Haar measure on
allows us to define the notion of
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
for (
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued) Borel functions defined on the group. In particular, one may consider various
''Lp'' spaces associated to the Haar measure μ. 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
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
. If
, then the Fourier transform is the function
on
defined by
where the integral is relative to
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, though ...
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 (the group and dual group) as follows (note that
is
Circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
):
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
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
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 (that is, since we can think about
as its own dual space we can ask for
to equal
) 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. (In effect it limits the
only to the exponent rather than as a pre-factor outside the integral sign.) 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 ''L''2 Fourier inversion theorems.
The group algebra
The space of integrable functions on a locally compact abelian group
is an
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, 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
In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element.
Definition
A right approxim ...
which is a net (or generalized sequence)
indexed on a directed set
such that
The Fourier transform takes convolution to multiplication, i.e. it is a homomorphism of abelian Banach algebras
(of norm ≤ 1):
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 (that is, not identically zero) multiplicative linear functionals on the group algebra; see section 34 of . This means the Fourier transform is a special case of the
Gelfand transform In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things:
* a way of representing commutative Banach algebras as algebras of continuous functions;
* the fact that for commutative C*-alg ...
.
Plancherel and ''L''2 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.
Since the complex-valued continuous functions of compact support on
are
-dense, there is a unique extension of the Fourier transform from that space to a
unitary operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the co ...
and we have the formula
Note that for non-compact locally compact groups
the space
does not contain
, so the Fourier transform of general
-functions on
is "not" given by any kind of integration formula (or really any explicit formula). To define the
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
Fourier transform. This is the content of the
Fourier inversion formula which follows.
In the case
the dual group
is naturally isomorphic to the group of integers
and the Fourier transform specializes to the computation of coefficients of
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
of periodic functions.
If
is a finite group, we recover the
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comp ...
. 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:
That
being compact implies
is discrete or that
being discrete implies that
is compact is an elementary consequence of the definition of the compact-open topology on
and does not need Pontryagin duality. One uses Pontryagin duality to prove the converses.
The
Bohr compactification is defined for any topological group
, regardless of whether
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''
of
is
, where ''H'' has the group structure
, but given the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
. Since the
inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iota ...
is continuous and a homomorphism, the dual morphism
is a morphism into a compact group which is easily shown to satisfy the requisite
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
.
Categorical considerations
Pontryagin duality can also profitably be considered
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
ially. In what follows, LCA is the
category of locally compact abelian groups and continuous group homomorphisms. The dual group construction of
is a contravariant functor LCA → LCA, represented (in the sense of
representable functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets an ...
s) by the circle group
as
In particular, the double dual functor
is ''covariant''.
A categorical formulation of Pontryagin duality then states that the
natural transformation between the identity functor on LCA and the double dual functor is an isomorphism. Unwinding the notion of a natural transformation, this means that the maps
are isomorphisms for any locally compact abelian group
, and these isomorphisms are functorial in
. This isomorphism is analogous to the
double dual
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of
finite-dimensional vector space
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to dist ...
s (a special case, for real and complex vector spaces).
An immediate consequence of this formulation is another common categorical formulation of Pontryagin duality: the dual group functor is an
equivalence of categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fro ...
from LCA to LCA
op.
The duality interchanges the subcategories of discrete groups and
compact groups. If
is a
ring and
is a left
–
module, the dual group
will become a right
–module; in this way we can also see that discrete left
–modules will be Pontryagin dual to compact right
–modules. The ring
of
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
s in LCA is changed by duality into its
opposite ring In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring is the ring ...
(change the multiplication to the other order). For example, if
is an infinite cyclic discrete group,
is a circle group: the former has
so this is true also of the latter.
Generalizations
Generalizations of Pontryagin duality are constructed in two main directions: for commutative
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
s that are not
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ...
, and for noncommutative topological groups. The theories in these two cases are very different.
Dualities for commutative topological groups
When
is a Hausdorff abelian topological group, the group
with the compact-open topology is a Hausdorff abelian topological group and the natural mapping from
to its double-dual
makes sense. If this mapping is an isomorphism, it is said that
satisfies Pontryagin duality (or that
is a ''reflexive group'', or a ''reflective group''). This has been extended in a number of directions beyond the case that
is locally compact.
In particular, Samuel Kaplan 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 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 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
-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 proved in 1995 that if
is a Hausdorff abelian topological group that satisfies Pontryagin duality, and the natural evaluation pairing
is (jointly) continuous, then
is locally compact. As a corollary, all non-locally compact examples of Pontryagin duality are groups where the pairing
is not (jointly) continuous.
Another way to generalize Pontryagin duality to wider classes of commutative topological groups is to endow the dual group
with a bit different topology, namely the ''topology of uniform convergence on
totally bounded set In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size� ...
s''. The groups satisfying the identity
under this assumption are called ''stereotype groups''. This class is also very wide (and it contains locally compact abelian groups), but it is narrower than the class of reflective groups.
Pontryagin duality for topological vector spaces
In 1952 Marianne F. Smith noticed that
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
s and
reflexive space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an is ...
s, being considered as topological groups (with the additive group operation), satisfy Pontryagin duality. Later B. S. Brudovskiĭ,
William C. Waterhouse and K. Brauner showed that this result can be extended to the class of all quasi-complete
barreled space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector.
A barrelled set or a b ...
s (in particular, to all
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to th ...
s). In the 1990s Sergei Akbarov gave a description of the class of the topological vector spaces that satisfy a stronger property than the classical Pontryagin reflexivity, namely, the identity
where
means the space of all linear continuous functionals
endowed with the ''topology of uniform convergence on totally bounded sets'' in
(and
means the dual to
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.
Dualities for non-commutative topological groups
For non-commutative locally compact groups
the classical Pontryagin construction stops working for various reasons, in particular, because the characters don't always separate the points of
, and the irreducible representations of
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
, and it is even not clear whether this set is a good choice for the role of the dual object for
. 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
was a Japanese mathematician who worked in algebraic number theory.
Biography
Tannaka was born in Matsuyama, Ehime Prefecture on December 27, 1908. After receiving a Bachelor of Science in mathematics from Tohoku Imperial University in 1932, he ...
(1938) and
Mark Krein
Mark Grigorievich Krein ( uk, Марко́ Григо́рович Крейн, russian: Марк Григо́рьевич Крейн; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet school of fu ...
(1949) constructed a duality theory for arbitrary compact groups known now as the
Tannaka–Krein duality. In this theory the dual object for a group
is not a group but a
category of its representations .
The theories of first type appeared later and the key example for them was the duality theory for finite groups. In this theory the category of finite groups is embedded by the operation
of taking
group algebra (over
) into the category of finite dimensional
Hopf algebra Hopf is a German surname. Notable people with the surname include:
* Eberhard Hopf (1902–1983), Austrian mathematician
* Hans Hopf (1916–1993), German tenor
* Heinz Hopf (1894–1971), German mathematician
* Heinz Hopf (actor) (1934–2001), Sw ...
s, so that the Pontryagin duality functor
turns into the operation
of taking the
dual vector space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
(which is a duality functor in the category of finite dimensional Hopf algebras).
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. From the 1980s the research in this area was resumed after the discovery of
quantum group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras ...
s, to which the constructed theories began to be actively transferred. These theories are formulated in the language of
C*-algebras
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
, or
Von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann algebra ...
s, and one of its variants is the recent theory of
locally compact quantum group In mathematics and theoretical physics, a locally compact quantum group is a relatively new C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf-algebra approaches. Earlier attempts at a uni ...
s.
One of the drawbacks of these general theories, however, is that in them the objects generalizing the concept of group are not
Hopf algebra Hopf is a German surname. Notable people with the surname include:
* Eberhard Hopf (1902–1983), Austrian mathematician
* Hans Hopf (1916–1993), German tenor
* Heinz Hopf (1894–1971), German mathematician
* Heinz Hopf (actor) (1934–2001), Sw ...
s in the usual algebraic sense. 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
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sho ...
of topological algebra.
See also
*
Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Pete ...
*
Cartier duality In mathematics,
Cartier duality is an analogue of Pontryagin duality for commutative group schemes. It was introduced by .
Definition using characters
Given any finite flat commutative group scheme ''G'' over ''S'', its Cartier dual is the group o ...
*
Stereotype space
Notes
References
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*{{cite journal, last=Waterhouse, first=William C., author-link=William C. Waterhouse, title=Dual groups of vector spaces, journal=
Pacific Journal of Mathematics
The Pacific Journal of Mathematics is a mathematics research journal supported by several universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisa ...
, year=1968, volume=26, issue=1, pages=193–196, doi=10.2140/pjm.1968.26.193, url=https://projecteuclid.org/euclid.pjm/1102986038, doi-access=free
Harmonic analysis
Duality theories
Theorems in analysis
Fourier analysis