| Non-commutative Harmonic Analysis |
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| CATEGORIES ABOUT NON-COMMUTATIVE HARMONIC ANALYSIS | |
| topological groups | |
| harmonic analysis | |
| representation theory | |
| duality theories | |
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In Mathematics , non-commutative harmonic analysis is the field in which results from Fourier Analysis are extended to Topological Group s which are not commutative. Since for Locally Compact Abelian Group s have a well-understood theory, Pontryagin Duality , which includes the basic structures of Fourier Series and Fourier Transform s, 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 Group s is understood, qualitatively and after the Peter-Weyl Theorem from the 1920s, as being generally analogous to that of Finite Group s and their Character Theory . 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 s, and also Algebraic Group s over P-adic Field s. These examples are of interest and frequently applied in Mathematical Physics , and contemporary Number Theory , particularly Automorphic Representation s. 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 ''L''2(''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 (see Discrete Series Representation ). For Semisimple Group s, and classes of Solvable Lie Group s, a very detailed theory is available. SEE ALSO |
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