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- Noncommutative_harmonic_analysis abstract "In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier transforms, the major business of non-commutative harmonic analysis is usually taken to be the extension of the theory to all groups G that are locally compact. The case of compact groups is understood, qualitatively and after the Peter–Weyl theorem from the 1920s, as being generally analogous to that of finite groups and their character theory.The main task is therefore the case of G that is locally compact, not compact and not commutative. The interesting examples include many Lie groups, and also algebraic groups over p-adic fields. These examples are of interest and frequently applied in mathematical physics, and contemporary number theory, particularly automorphic representations.What to expect is known as the result of basic work of John von Neumann. He showed that if the von Neumann group algebra of G is of type I, then L2(G) as a unitary representation of G is a direct integral 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.".
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- Noncommutative_harmonic_analysis wikiPageRevisionID "690310722".
- Noncommutative_harmonic_analysis wikiPageWikiLink Algebraic_group.
- Noncommutative_harmonic_analysis wikiPageWikiLink Automorphic_form.
- Noncommutative_harmonic_analysis wikiPageWikiLink Category:Duality_theories.
- Noncommutative_harmonic_analysis wikiPageWikiLink Category:Harmonic_analysis.
- Noncommutative_harmonic_analysis wikiPageWikiLink Category:Topological_groups.
- Noncommutative_harmonic_analysis wikiPageWikiLink Character_theory.
- Noncommutative_harmonic_analysis wikiPageWikiLink Commutative_property.
- Noncommutative_harmonic_analysis wikiPageWikiLink Compact_group.
- Noncommutative_harmonic_analysis wikiPageWikiLink Direct_integral.
- Noncommutative_harmonic_analysis wikiPageWikiLink Discrete_series_representation.
- Noncommutative_harmonic_analysis wikiPageWikiLink Finite_group.
- Noncommutative_harmonic_analysis wikiPageWikiLink Fourier_analysis.
- Noncommutative_harmonic_analysis wikiPageWikiLink Fourier_series.
- Noncommutative_harmonic_analysis wikiPageWikiLink Fourier_transform.
- Noncommutative_harmonic_analysis wikiPageWikiLink Harmonic_analysis.
- Noncommutative_harmonic_analysis wikiPageWikiLink John_von_Neumann.
- Noncommutative_harmonic_analysis wikiPageWikiLink Langlands_program.
- Noncommutative_harmonic_analysis wikiPageWikiLink Lie_group.
- Noncommutative_harmonic_analysis wikiPageWikiLink Locally_compact_space.
- Noncommutative_harmonic_analysis wikiPageWikiLink Mathematical_physics.
- Noncommutative_harmonic_analysis wikiPageWikiLink Mathematics.
- Noncommutative_harmonic_analysis wikiPageWikiLink Number_theory.
- Noncommutative_harmonic_analysis wikiPageWikiLink Orbit_method.
- Noncommutative_harmonic_analysis wikiPageWikiLink P-adic_number.
- Noncommutative_harmonic_analysis wikiPageWikiLink Peter–Weyl_theorem.
- Noncommutative_harmonic_analysis wikiPageWikiLink Plancherel_theorem.
- Noncommutative_harmonic_analysis wikiPageWikiLink Pontryagin_duality.
- Noncommutative_harmonic_analysis wikiPageWikiLink Selberg_trace_formula.
- Noncommutative_harmonic_analysis wikiPageWikiLink Semisimple_algebraic_group.
- Noncommutative_harmonic_analysis wikiPageWikiLink Solvable_Lie_algebra.
- Noncommutative_harmonic_analysis wikiPageWikiLink Spectrum_of_a_C*-algebra.
- Noncommutative_harmonic_analysis wikiPageWikiLink Topological_group.
- Noncommutative_harmonic_analysis wikiPageWikiLink Unitary_representation.
- Noncommutative_harmonic_analysis wikiPageWikiLink Von_Neumann_algebra.
- Noncommutative_harmonic_analysis wikiPageWikiLink Zonal_spherical_function.
- Noncommutative_harmonic_analysis wikiPageWikiLinkText "Noncommutative harmonic analysis".
- Noncommutative_harmonic_analysis wikiPageWikiLinkText "noncommutative harmonic analysis".
- Noncommutative_harmonic_analysis wikiPageUsesTemplate Template:Reflist.
- Noncommutative_harmonic_analysis subject Category:Duality_theories.
- Noncommutative_harmonic_analysis subject Category:Harmonic_analysis.
- Noncommutative_harmonic_analysis subject Category:Topological_groups.
- Noncommutative_harmonic_analysis hypernym Field.
- Noncommutative_harmonic_analysis type Space.
- Noncommutative_harmonic_analysis type Theory.
- Noncommutative_harmonic_analysis comment "In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier transforms, the major business of non-commutative harmonic analysis is usually taken to be the extension of the theory to all groups G that are locally compact.".
- Noncommutative_harmonic_analysis label "Noncommutative harmonic analysis".
- Noncommutative_harmonic_analysis sameAs Q7049222.
- Noncommutative_harmonic_analysis sameAs Analyse_harmonique_non_commutative.
- Noncommutative_harmonic_analysis sameAs 非可換調和解析.
- Noncommutative_harmonic_analysis sameAs m.025xqyy.
- Noncommutative_harmonic_analysis sameAs Q7049222.
- Noncommutative_harmonic_analysis wasDerivedFrom Noncommutative_harmonic_analysis?oldid=690310722.
- Noncommutative_harmonic_analysis isPrimaryTopicOf Noncommutative_harmonic_analysis.