Math 2.0

Abstract harmonic analysis Definition

Noncommutative harmonic analysis

Builds on Harmonic analysis on groupsLocally compact groupHaar measureUnitary representationLeft regular representation

Definition

Let GG be a locally compact group, not assumed abelian, with a left Haar measure. Noncommutative harmonic analysis is the part of harmonic analysis on groups that studies GG through its irreducible unitary representations. Its main questions are:

  1. how the left regular representation of GG on L2(G)L^2(G) decomposes as a direct sum or direct integral of irreducible unitary representations;
  2. how a function ff on GG is recovered from the operators π(f)=∫Gf(x)π(x) dx\pi(f) = \int_G f(x) \pi(x) \, dx, where π\pi runs over the irreducible unitary representations;
  3. how the L2L^2 norm of ff is expressed through these operators, by a Plancherel formula.

Remarks

Example: for a compact group GG, the Peter–Weyl theorem answers question 1. The regular representation is the Hilbert direct sum of the irreducible representations π\pi, each with multiplicity dim⁡π\dim \pi. For a finite group of order nn this gives the identity n=∑π(dim⁡π)2n = \sum_\pi (\dim \pi)^2.

Non-example: for GG abelian, every irreducible unitary representation is one-dimensional, given by a character. The theory is then Fourier analysis on the dual group, which is the commutative case and not what the term refers to.

The term abstract harmonic analysis is often used for the abelian and the general case together.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Haar measure, concept, no evidence levelHaar measureconceptHarmonic analysis on groups, concept, no evidence levelHarmonic analysis on …conceptLeft regular representation, concept, no evidence levelLeft regular represen…conceptLocally compact group, concept, no evidence levelLocally compact groupconceptUnitary representation, concept, no evidence levelUnitary representationconceptNoncommutative harmonic analysis, definition, no evidence levelNoncommutative harmon…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Haar measure concept
  2. Harmonic analysis on groups concept
  3. Left regular representation concept
  4. Locally compact group concept
  5. Unitary representation concept
  6. Noncommutative harmonic analysis definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Noncommutative_harmonic_analysis
About this page
TagCWTY64YTYN
URIm2://CWTY64YTYN
Kinddefinition
WritingW0 stub
MSC43A65, 43A30, 22D10
DefinesNoncommutative harmonic analysis
UsesHarmonic analysis on groups, Locally compact group, Haar measure, Unitary representation, Left regular representation
Changed3 November 2026

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