SUBJECT

Title

Topological vector spaces and Banach-algebras

Type of instruction

lecture + practical

Level

master

Part of degree program
Credits

3+3

Recommended in

Semesters 1-4

Typically offered in

Autumn/Spring semester

Course description

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.

Readings
  • 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