Operator Algebras - M2 - 8EC

Prerequisites:
Prerequisites are the standard algebra and analysis courses and a good background in general topology and functional analysis (FA). Preferably from the Mastermath course Functional Analysis, but a substantial introduction to FA from your university might suffice. At some point, 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.
(The material in the appendix of Murphy's book is quite sufficient.)

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 have been teaching at Nijmegen since 2020. You can find them at http://www.math.ru.nl/~mueger/functionalanalysis.pdf
I expect you to be fairly familiar with the material in Appendix A and Part I of these notes as well as the operator theory in Part II, but I will cover some of the less standard material in the lectures.

In order to limit the overlap with the MasterMath course on functional analysis, I will not cover the basic theory of unital Banach algebras in detail! I expect you to have met the material in Chapter I of Murphy's book (as specialized to unital algebras) or in Sections 11 and 17.1 of my lecture notes.

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:

  • inductive limits of C*-algebras (UHF and AF algebras)
  • some interesting examples of C*-algebras

Lecturer:
Michael Müger (RU)