Noncommutative harmonic analysis
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...

s which are not commutative. Since for locally compact abelian groups have a well-understood theory, Pontryagin duality
Pontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...

, which includes the basic structures of Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

 and Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

s, the major business of non-commutative harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

 is usually taken to be the extension of the theory to all groups G that are locally compact. The case of compact group
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...

s is understood, qualitatively and after the Peter–Weyl theorem from the 1920s, as being generally analogous to that of finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

s and their character theory
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....

.

The main task is therefore the case of G which is locally compact, not compact and not commutative. The interesting examples include many Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

s, and also algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

s over p-adic fields. These examples are of interest and frequently applied in mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

, and contemporary number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, particularly automorphic representations.

What to expect is known as the result of basic work of John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

. He showed that if the von Neumann group algebra of G is of type I, then L2(G) as a unitary representation
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 is a unitary operator for every g ∈ G...

 of G is a direct integral
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 in 1949 by John von Neumann in one of the papers...

 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
Plancherel theorem
In mathematics, the Plancherel theorem is a result in harmonic analysis, proved by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum....

 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
Dual group
In mathematics, the dual group may be:* The Pontryagin dual of a locally compact abelian group* The Langlands dual of a reductive algebraic group* The Deligne-Lusztig dual of a reductive group over a finite field....

 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.

See also

  • Selberg trace formula
    Selberg trace formula
    In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of G on the space L2 of square-integrable functions, where G is a Lie group and Γ a cofinite discrete group...

  • Langlands program
    Langlands program
    The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....

  • Kirillov orbit theory
    Kirillov orbit theory
    In mathematics, the orbit method establishes a correspondence between irreducible unitary representations of a Lie group and its coadjoint orbits: orbits of the action of the group on the dual space of its Lie algebra...

  • Discrete series representation
    Discrete series representation
    In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²...

  • Zonal spherical function
    Zonal spherical function
    In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G...

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