Topg - Topological Groups Part III, Spring 2008 T. W....

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Unformatted text preview: Topological Groups Part III, Spring 2008 T. W. Korner March 8, 2008 Small print This is just a first draft for the course. The content of the course will be what I say, not what these notes say. Experience shows that skeleton notes (at least when I write them) are very error prone so use these notes with care. I should very much appreciate being told of any corrections or possible improvements and might even part with a small reward to the first finder of particular errors. This document is written in L A T E X2e and available in tex, dvi, ps and pdf form from my home page http://www.dpmms.cam.ac.uk/~twk/ . My e-mail address is twk@dpmms.cam.ac.uk . In the middle of the 20th century it was realised that classical Fourier Analysis could be extended to locally compact Hausdorff Abelian groups. The object of this course (which may not be completely achieved) is to show how this is done. (Specifically we wish to get as far as the first two chapters of the book of Rudin [6].) The main topics will thus be topological groups in general, Haar measure, Fourier Analysis on locally compact Hausdorff Abelian groups, Pontryagin duality and the principal structure theorem. Although we will not need deep results, we will use elementary functional analysis, measure theory and the elementary theory of commutative Banach algebras. (If you know two out of three you should have no problems, if only one out of three then the course is probably a bridge too far.) Preliminary reading is not expected but the book by Deitmar [1] is a good introduction. Contents 1 Prelude 2 2 Topological groups 5 3 Subgroups and quotients 7 4 Products 9 5 Metrisability 12 6 The Haar integral 14 1 7 Existence of the Haar integral 17 8 The space L 1 ( G ) 19 9 Characters 21 10 Fourier transforms of measures 22 11 Discussion of the inversion theorem 23 12 The inversion theorem 25 13 Pontryagin duality 26 14 Structure theorems 27 15 Final remarks 29 1 Prelude We start with the following observations 1 . Lemma 1.1. Consider R , T = R / Z , S 1 = { C : | | = 1 } and Z with their usual (Euclidean) metrics. Then R , T and Z are Abelian groups under addition and S 1 is a group under multiplication. (i) The continuous homomorphisms : R S 1 are precisely the maps a ( t ) = exp( iat ) [ t R ] with a R . (ii) The continuous homomorphisms : T S 1 are precisely the maps a ( t ) = exp(2 iat ) [ t T ] with a Z . (iii) The continuous homomorphisms : Z S 1 are precisely the maps a ( t ) = exp(2 iat ) [ t Z ] with a T . Exercise 1.2. We use the notation of Lemma 1.1. (i) Show that the non-zero Borel measures on R such that integraldisplay R f ( x y ) d ( x ) = integraldisplay R f ( x ) d ( x ) for all continuous functions f : R R of compact support and all y R are precisely the non-zero multiples of m the Lebesgue measure on R . In other 1 I shall assume a lot of notation and results, not because I assume that my audience...
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Topg - Topological Groups Part III, Spring 2008 T. W....

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