- Lecturer Prof. Dr. Marc Burger
- Coordinator Victor Jaeck

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.

- Banach algebras and the spectral radius formula,
- Guelfand's theory of abelian Banach algebras,
- Locally compact groups, Haar measure, properties of the convolution product,
- Locally compact abelian groups, the dual group, basic properties of the Fourier transform,
- Positive definite functions and Bochner's theorem,
- The Fourier inversion formula, Plancherel's theorem,
- Pontryagin duality and consequences,
- 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,
- Guelfand's theory of abelian C*-algebras and applications to the spectral theorem for normal operators,
- Property (T).

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 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.

Time | Room | Assistant | Language |
---|---|---|---|

Mo 09:15-10 | HG E 33.3 | Valentin Bosshard | English |

Mo 09:15-10 | HG F 26.5 | Cynthia Bortolotto | English |

- 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

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