Functional Analysis II Spring 2022

Lecturer

Prof. Dr. Marc Burger

Coordinator

Victor Jaeck

Content

The course will focus essentially on the theory of abelian Banach algebras and its applications to harmonic analysis on locally compact abelian groups, and spectral theorems. Time permitting we will talk about a fundamental property of highly non abelian groups, namely property (T); one of the spectacular applications thereof is the explicit construction of expander graphs.

Syllabus

1. Banach algebras and the spectral radius formula,
2. Guelfand's theory of abelian Banach algebras,
3. Locally compact groups, Haar measure, properties of the convolution product,
4. Locally compact abelian groups, the dual group, basic properties of the Fourier transform,
5. Positive definite functions and Bochner's theorem,
6. The Fourier inversion formula, Plancherel's theorem,
7. Pontryagin duality and consequences,
8. Regular abelian Banach algebras, minimal ideals and Wiener's theorem for general locally compact abelian groups. Applications to Wiener-Ikehara and the prime number theorem,
9. Guelfand's theory of abelian C*-algebras and applications to the spectral theorem for normal operators,
10. Property (T).

Lecture notes