Notes on Abstract Harmonic Analysis
by Seung-Hyeok Kye
RIM-GARC Lecture Notes Ser. No. 20, Seoul National University, 1994, pp.100

This is the collection of notes which have been distributed during the lectures on abstract harmonic analysis in the fall semester of the academic year 1993 at Seoul National University. The main topic of the lecture was to introduce measure theoretic or functional analysis approach to the group representation theory. It has been assumed that the audience has good backgrounds on abstract measure theory and elementary functional analysis with Hahn-Banach and Banach-Steinhaus Theorems. Some advanced functional analysis techniques such as Banach-Alaoglu, Krein-Milman, Stone-Weierstrass Theorems and the spectral decomposition theorem have been discussed briefly during the course.

One of the breakthroughs in the group representation theory was H. Weyl's observation that the multiplication of the group ring is nothing but the convolution in Fourier analysis. This observation leads him to study the representations of compact groups, generalizing those of finite groups. The existence of left invariant measure for an arbitrary locally compact group by Haar enables us to define the convolution and involution on the Banach space $L^1(G)$, to get a Banach $*$-algebra. We begin this note with the proof of the existence and uniqueness of the Haar measure, and examine elementary properties of the convolution and involution. Every unitary representation of a group $G$ naturally induces a $*$-representation of the Banach $*$-algebra $L^1(G)$, where positive linear functional plays crucial roles. We conclude Chapter I with elementary properties of positive linear functionals on $L^1(G)$, or equivalently positive definite functions on $G$.

In Chapter II, we exclusively consider locally compact abelian (LCA) groups, whose representation theory amounts to the Fourier transform which converts $L^1$-functions (respectively complex regular Borel measure) on $G$ to continuous functions on the dual group $\widehat G$ consisting of characters which vanish at infinity (respectively which is bounded). The Fourier transform on $L^1(G)$ may be generalized to the Gelfand transform on arbitrary commutative Banach algebras. Classical inversion formula for periodic functions and Plancherel transform on the real line will be proved for arbitrary LCA groups, using Bochner Theorem on positive definite functions. The central theme Chapter II is the Pontryagin duality, which says that the double dual of an LCA group is topologically isomorphic to the original group. With this duality in hands, every result in Fourier transform has the dual interpretation. The critical point of the Pontryagin duality is that there are sufficiently many characters, and this will be proved using the inversion formula. The Fourier transform may be extended to the distribution if the involving group has the differential structures. We restrict ourselves to the circle groups, and study the Fourier-Schwartz transform of periodic distributions. The range of this transform covers every slowly increasing sequence, and so every trigonometric series slowly increasing coefficients defines a distribution in a suitable sense. This completes the idea of Jean Baptiste Joseph Fourier that every periodic function is represented by a trigonometric series.

In the case of non-abelian groups, the characters should be replaced by irreducible representations. We begin Chapter III with the establishment of the correspondences between continuous unitary representations on $G$, non-degenerated $*$-representations of $L^1(G)$ and continuous positive definite functions on $G$. Irreducible representations correspond to continuous positive definite functions which are extreme in a sense. Employing functional analysis techniques such as Banach-Alaoglu and Krein-Milman Theorems, we show that there are sufficiently many irreducible representations for an arbitrary locally compact Hausdorff group. We will pay attention to compact groups, for which every irreducible representation is finite-dimensional. We decompose the regular representation of a compact group into irreducible representations, which amounts to the Fourier series expansion for periodic functions. Compact groups also enjoy dualities, and we discuss here the classical Tannaka-Krein duality. We close this note by finding out all irreducible representations for the simplest non-abelian compact groups such as special unitary and orthogonal groups with low dimensions.

The lack of time prevents us to continue our study on general locally compact groups. The author hopes to continue this part in an another chance. The author would like to express his deep gratitude to all participants of the lecture. Another special thanks are due to Professors Jaihan Yoon, Doham Kim, Hong-Jong Kim and Insok Lee. Discussions with them together with their comments were indispensable to prepare this note.

1. Haar Integrals
2. Convolutions
3. Positive Definite Functions

4. Dual Groups and the Fourier Transforms
5. Gelfand Transforms
6. Inversion Formula and the Plancherel Transforms
7. Pontryagin-van Kampen Duality
8. Smoothness

9. Unitary Representations
10. Irreducible Representations
11. Compact Groups
12. Tannaka-Krein Duality
13. The Groups SU(2) and SO(3)