The students are familiar with the basics of C*-algebras and von Neumann algebras, allowing them to specialize further or to apply operator algebras in the context of non-commutative geometry or the theory of infinite quantum systems.

In 1929, J. von Neumann began studying what came to be called von Neumann algebras. C*-algebras were introduced by Gelfand and Nalmark in 1943. These two subjects together form the discipline of Operator Algebras, an important part of Functional Analysis with many applications in harmonic analysis and representation theory, quantum group theory, Connes' non-commutative geometry, and mathematical physics (quantum mechanics and field theory, statistical physics). The aim of this course is to lay the foundations for further studies of the subject and its applications.

We will cover at least the following subjects:

Banach algebras, in particular spectral theory

commutative C*-algebras

ideals, quotients, homomorphisms

states and representations

weak topologies, density theorems

von Neumann algebras

Possible further subjects, time permitting:

tensor products of C*-algebras

some interesting examples of C*-algebras

projections in von Neumann algebras and the type classification

**Prerequisites**

Standard Bachelor courses on Analysis, plus some knowledge in Functional Analysis, as provided by the Mastermath course Functional Analysis (vGaans) or at least an introductory course like the "Introduction to functional analysis" as taught in Nijmegen.

As an indication, you should be familiar with a reasonable part of the material in the first three chapters of Gert Pedersen's book "Analysis NOW", Springer-Verlag, 1989.

- Docent: Martijn Caspers
- Docent: Michael Mueger