**Course contents**

• Basic functional analysis, Banach spaces, linear operators

• Banach algebras, spectrum, Gelfand transform, (holomorphic) functional calculus

• C ∗ -algebras and their basic properties

• Gelfand-Naimark duality between C ∗ -algebras and locally compact Hausdorff spaces

• Continuous functional calculus, positive elements, approximate units

• Basics of von Neumann algebras

• (pure) states and (irreducible) representations

• Gelfand-Naimark-Segal (GNS) construction

Further possible topics (if time permits)

• multipliers

• tensor products

• group C ∗ -algebras, crossed products

**Examination**

The exam will consist of a seminar presentation on an advanced topic related to the

course.

**Recommended literature**

1. B. Blackadar, Operator algebras: Theory of C ∗ -algebras and von Neumann alge-

bras, Encyclopaedia of Mathematical Sciences, vol. 13, Springer, 2006.

2. J. Conway, A course in functional analysis Second edition. Graduate Texts in

Mathematics, 96. Springer-Verlag, New York, 1990.