SUBJECT
Topological vector spaces and Banach-algebras
lecture + practical
master
3+3
Semesters 1-4
Autumn/Spring semester
Basic properties of linear topologies. Initial linear topologies. Locally compact topological vector spaces. Metrisable topological vector spaces. Locally convex and polinormed spaces. Inductive limit of locally convex spaces. Krein-Milmans theorem. Geometric form of Hahn-Banach theorem and separation theorems. Bounded sets in topological vector spaces. Locally convex function spaces. Ascoli theorems. Alaoglu-Bourbaki theorem. Banach-Alaoglu theorem. Banach-Steinhaus theorem. Elementary duality theory. Locally convex topologies compatible with duality. Mackey-Arens theorem. Barrelled, bornologic, reflexive and Montel-spaces. Spectrum in a Banach-algbera. Gelfand-representation of a commutative complex Banach-algebra. Banach-*-algebras and C*-algebras. Commutative C*-algebras (I. Gelgand-Naimark theorem). Continuous functional calculus. Universal covering C*-algebra and abstract Stone’s theorem. Positive elements in C*-algebras. Baer C*-algebras.
- N. Bourbaki: Espaces vectoriels topologiques, Springer, Berlin-Heidelberg-New York, 2007
- N. Bourbaki: Théories spectrales, Hermann, Paris, 1967
- J. Dixmier: Les C*-algébres et leurs représentations, Gauthier-Villars Éd., Paris, 1969