Prerequisites are the standard algebra and analysis courses and a good background in general topology and functional analysis. Preferably from the Mastermath course Functional Analysis, but a good introduction to FA from your university might suffice. However, we will need some of the theory of locally convex spaces, so that you will have to do some studying on your own if you haven't met them before.

As to the other material from FA, you can get an idea of what I expect from the lecture notes on the 3rd year course that I'm teaching at Nijmegen in the Fall 2020. You can find them at

I do not expect that you have met all the material on Banach and C*-algebras contained in these notes, but I will assume known all of Appendix A, Part I and everything in Part II that concerns operator theory and weak(-*) topologies.

Aim of the course

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


Michael Müger (RU)