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

Date Notes Comments
Monday 21.02.22 Lecture 1 We have not treated Examples 1.9. (3), (4) and (7).
Thursday 24.02.22 Lecture 2
Monday 28.02.22 Lecture 3, Lecture 3 bis Lecture 3 bis is page 13 of the course.
Thursday 03.03.22 Lecture 4
Monday 07.03.22 Lecture 5 Prop. 3.11: for the last conclusion, it is necessary to request that the norm of the unit is one.
Thursday 10.03.22 Lecture 6, Lecture 6 appendix
Monday 14.03.22 Lecture 7
Thursday 17.03.22 Lecture 8
Monday 21.03.22 Lecture 9
Thursday 24.03.22 Lecture 10
Monday 28.03.22 Lecture 11
Thursday 31.03.22 Lecture 12
Monday 04.04.22 Lecture 13
Thursday 07.04.22 Lecture 14
Monday 11.04.22 Lecture 15
Thursday 14.04.22 Lecture 16
Monday 25.04.22 Lecture 17
Monday 02.05.22 Lecture 18 There was no class on Thursday 28.04.
Thursday 14.04.22 Lecture 19
Monday 09.05.22 Lecture 20
Thursday 12.05.22 Lecture 21
Monday 16.05.22 Lecture 22
Thursday 19.05.22 Lecture 23
Monday 23.05.22 Lecture 24
Monday 30.05.22 Lecture 25
Thursday 02.06.22 Lecture 26

Exercise Classes

Here is the SAM-Up web page where you can submit your exercises for correction: SAM-Up

Exercise sheet Due by
Exercise sheet 1 04.03.2022
Exercise sheet 2 11.03.2022
Exercise sheet 3 18.03.2022
Exercise sheet 4 25.03.2022
Exercise sheet 5 01.04.2022
Exercise sheet 6 08.04.2022
Exercise sheet 7 22.04.2022
Exercise sheet 8 29.04.2022
Exercise sheet 9 06.05.2022
Exercise sheet 10 13.05.2022
Exercise sheet 11 20.05.2022
Exercise sheet 12 23.05.2022
Exercise sheet 13 03.06.2022
Hjalti Isleifsson wrote some possible solutions for the exercise sheets: Few solutions to the exercises.

TimeRoomAssistantLanguage
Mo 09:15-10 HG E 33.3 Valentin Bosshard English
Mo 09:15-10 HG F 26.5 Cynthia Bortolotto English

Bibliography

  • M.Einsiedler, T. Ward: Functional Analysis, Spectral Theory, and Applications, GTM Springer, 2017
  • I. Gelfand, D. Raikov, G. Shilov: Commutative Normed Rings, Chelsea 1964
  • E. Kaniuth: A Course in Commutative Banach Algebras, GTM Springer, 2009
  • W. Rudin: Fourier Analysis on Groups, Dover, 1967
  • M. Takesaki: Theory of Operator Algebras, Springer, 1979

Prerequisites

  • Point set topology,
  • Basic measure theory,
  • Basics of functional analysis specifically: Banach-Steinhaus, Banach-Alaoglu, and Hahn-Banach.