A Course in Abstract Harmonic Analysis
Abstract theory remains an indispensable foundation for the study of concrete cases. It shows what the general picture should look like and provides results that are useful again and again. Despite this, however, there are few, if any introductory texts that present a unified picture of the general abstract theory.
A Course in Abstract Harmonic Analysis offers a concise, readable introduction to Fourier analysis on groups and unitary representation theory. After a brief review of the relevant parts of Banach algebra theory and spectral theory, the book proceeds to the basic facts about locally compact groups, Haar measure, and unitary representations, including the Gelfand-Raikov existence theorem. The author devotes two chapters to analysis on Abelian groups and compact groups, then explores induced representations, featuring the imprimitivity theorem and its applications. The book concludes with an informal discussion of some further aspects of the representation theory of non-compact, non-Abelian groups.
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Abelian action acts algebra analysis apply associated Banach algebra Banach space Borel bounded called Cc(G character clearly closed commutes complete condition consider construction contains continuous converges Corollary correspondence countable decomposition define definition denote dense determined direct sum discrete easily easy element equivalent example exists fact factor field Finally finite follows formula Fourier transform functions give given group G Haar measure hence Hilbert space ideal identify identity implies induced inner integral invariant inverse irreducible representations isomorphism Lemma linear locally compact group Moreover multiplication namely neighborhood norm obtain operator orbit orthogonal particular Plancherel measure positive positive type projection Proof Proposition prove range regular representation of G respect result satisfies spectral subgroup subset subspace supp Suppose Theorem theory topology translations trivial unique unit unitary representation vector weak