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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Banach algebra Borel set C+(G CC(G closed subgroup commutes compact Hausdorff space continuous function converges Corollary coset cyclic decomposition define denote dense direct integral direct sum example finite follows Fourier transform functions of positive G is compact Gelfand transform group G Haar measure Hausdorff space hence Hilbert space homomorphism identify inner product invariant inverse irreducible representations isometry isomorphism Jg/h L(S x G left Haar measure Let G Lie groups linear functional linear span Ll(G locally compact group locally compact Hausdorff Mackey measure on G Moreover multiplication neighborhood nonzero norm open set operator orbit orthogonal orthonormal basis positive type projection-valued measure Proof Proposition prove pseudomeasure quasi-invariant measure Radon measure regular representation representation of G result Schur's lemma second countable spectral theorem subalgebra subset subspace supp Suppose G system of imprimitivity theory topology trace-class trivial unique unitary equivalence unitary representation vector fields