SUBJECT

Title

Representations of Banach-*-algebras and abstract harmonic analysis

Type of instruction

lecture + practical

Level

master

Faculty

Faculty of Science

Part of degree program

Mathematics MSc

Credits

2+2

Recommended in

Semesters 1-4

Typically offered in

Autumn/Spring semester

Course description

Representations of *-algebras. Positive functionals and GNS-construction. Representations of Banach-*-algebras. Gelfand-Raikoff theorem. The second Gelfand-Naimark theorem. Hilbert-integral of representations. Spectral theorems for C*-algebras and measurable functional calculus. Basic properties of topological groups. Continuous topological and unitary representations. Radon measures on locally compact spaces. Existence and uniqueness of left Haar-measure on locally compact groups. The modular function of a locally compact group. Regular representations. The group algebra of a locally compact group. The main theorem of abstract harmonic analysis. Gelfand-Raikoff theorem. Unitary representations of compact groups (Peter-Weyl theorems). Unitary representations of commutative locally compact groups (Stone-theorems). Factorization of Radon measures. Induced unitary representations (Mackey-theorems).

Readings

J. Dixmier: Les C*-algébres et leurs représentations, Gauthier-Villars Éd., Paris, 1969
E.Hewitt-K.Ross: Abstract Harmonic Analysis, Vols I-II, Springer-Verlag, 1963-1970