Mathematics

This course provides a brief introduction to C*-algebras. The course will cover at least the following topics:

- brief recap on Banach spaces and linear operators;

- Banach algebras: spectrum, Gelfand transform, Stone-Weierstrass theorem;

- C*-algebras: definition and basic properties;

- Gelfand-Naimark duality between commutative C*-algebras and locally compact Hausdorff spaces;

- continuous functional calculus;

- positive elements, approximate units;

- von Neumann algebras: definition and basic properties;

- (pure) states and (irreducible) representations;

- GNS (Gelfand-Naimark-Segal) construction.