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Functional Analysis II
Last Updated: 2026-02-05 16:37:35
Abstract
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.
Objective
Acquire fluency with abelian Banach algebras in order to apply their theory to harmonic analysis on locally compact groups and to spectral theorems
Content
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).
Resources
Lecture Notes
https://drive.google.com/file/d/1SlFesBUfhI5BvUyb_htHKZjLZ8gpDKOG/view
Literature
[EiWa] M. Einsiedler and T. Ward. Functional analysis, spectral theory, and applications. Vol. 276. Graduate Texts in Mathematics. Springer, 2017. [GeRaSh] I. Gelfand, D. Raikov, and G. Shilov. Commutative normed rings. Translated from the Russian, with a supplementary chapter. Chelsea Publishing Co., New York, 1964. [Ka] E. Kaniuth. A Course in Commutative Banach Algebras. Vol. 246. Graduate Texts in Mathematics. Springer, New York, 2009. [RaVa] D. Ramakrishnan and R. J. Valenza. Fourier Analysis on Number Fields. Vol. 186.Graduate Texts in Mathematics. Springer, New York, 1999. [Ru1] W. Rudin. Fourier Analysis on Groups. Interscience Tracts in Pure and Applied Mathematics 12. Interscience Publishers, New York-London, 1962. [Ru2] W. Rudin. Functional Analysis. Second Edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. [Ru3] W. Rudin. Real and Complex Analysis. Third Edition. McGraw-Hill Book Co., New York, 1987. [Ta] M. Takesaki. Theory of Operator Algebras I. Springer, New York, 1979. [We] A. Weil. Basic Number Theory. Classics in Mathematics. Reprint of the second (1973) edition. Springer Berlin, Heidelberg, 1995. [Zi] R. J. Zimmer. Essential Results of Functional Analysis. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1990.
General Information
- Language
- English
- Levels
- BSC , MSC
- Frequency
- Yearly recurring
Examination
- Type
- session examination
- Mode
- oral 30 minutes
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture | Functional Analysis II |
|
4 h weekly |
| exercise |
Functional Analysis II
Groups are selected in myStudies.
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|
1 h weekly |
Offered In
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Core Courses (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
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