mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, noncommutative harmonic analysis is the field in which results from

Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...

are extended to

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 st ...

s that are not

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

. Since locally compact abelian groups have a well-understood theory,

Pontryagin duality In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...

, which includes the basic structures 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 '' ...

and

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 ...

s, the major business of non-commutative

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...

is usually taken to be the extension of the theory to all groups ''G'' that are

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 ev ...

. The case of

compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...

s is understood, qualitatively and after the

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 Peter, ...

from the 1920s, as being generally analogous to that of

finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or ma ...

s and their

character theory In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information ab ...

. The main task is therefore the case of ''G'' that is locally compact, not compact and not commutative. The interesting examples include many

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...

s, and also

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...

s over

p-adic field In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extens ...

s. These examples are of interest and frequently applied in

mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...

, and contemporary

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

, particularly

automorphic representation In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset ...

s. What to expect is known as the result of basic work of

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 c ...

. He showed that if the von Neumann group algebra of ''G'' is of type I, then ''L''²(''G'') as a

unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...

of ''G'' is a

direct integral In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced i ...

of irreducible representations. It is parametrized therefore by the unitary dual, the set of isomorphism classes of such representations, which is given the hull-kernel topology. The analogue of the Plancherel theorem is abstractly given by identifying a measure on the unitary dual, the Plancherel measure, with respect to which the direct integral is taken. (For Pontryagin duality the Plancherel measure is some Haar measure on the dual group to ''G'', the only issue therefore being its normalization.) For general locally compact groups, or even countable discrete groups, the von Neumann group algebra need not be of type I and the regular representation of ''G'' cannot be written in terms of irreducible representations, even though it is unitary and completely reducible. An example where this happens is the infinite symmetric group, where the von Neumann group algebra is the hyperfinite type II₁ factor. The further theory divides up the Plancherel measure into a discrete and a continuous part. For semisimple groups, and classes of solvable Lie groups, a very detailed theory is available.

References

*"Noncommutative harmonic analysis: in honor of Jacques Carmona", Jacques Carmona, Patrick Delorme, Michèle Vergne; Publisher Springer, 2004 ''Noncommutaive Harmonic Analysis: In Honor of Jacques Carmona
/ref> * Yurii I. Lyubich. ''Introduction to the Theory of Banach Representations of Groups''. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988.

Notes

{{reflist Topological groups * Duality theories

See also

References

Notes