A Course in Abstract Harmonic AnalysisAbstract 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. |
Contents
Banach Algebras and Spectral Theory | 1 |
Locally Compact Groups | 34 |
Basic Representation Theory | 71 |
Analysis on Locally Compact Abelian Groups | 87 |
Analysis on Compact Groups | 125 |
31 | 140 |
43 | 147 |
Induced Representations | 151 |
Further Topics in Representation Theory | 201 |
67 | 208 |
87 | 221 |
Appendices | 253 |
| 263 | |
| 267 | |
| 270 | |
| 276 | |
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Common terms and phrases
Abelian group antilinear Banach algebra Cc(G Ce(G closed subgroup commutes compact Hausdorff space continuous function converges Corollary cyclic decomposition define denote dense direct integral direct sum Dixmier 29 dµ(a equivalence class example field of Hilbert finite follows Fourier transform functions of positive ƒ ƒ G₁ group G H₁ H₂ Haar measure hence Hilbert space homomorphism identify ind(o induced representation inner product invariant inverse irreducible representations isometry isomorphism L¹(G L²(G Let G Lie groups linear span locally compact group Mackey measurable field Moreover multiplication neighborhood norm operator orbit orthonormal basis positive type Prim(G projection-valued measure Proof Proposition prove pseudomeasure Radon measure regular representation representation of G result Schur's lemma second countable spectral theorem subset subspace supp Suppose G system of imprimitivity tensor product theory trace-class trivial unique unitary equivalence unitary representation vector fields ΔΗ(ξ