### M1: Operator Algebras - 8EC

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.

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

Rules about Homework / Exam

If your exam grade is < 5 then this is your end grade. Otherwise it is the maximum of the following two numbers: (1) your exam grade, (2) the average of your exam grade (75%) and your grade for the take-home exercises (25%).

Lecture Notes / Literature

Necessary

We will mainly use the following book: Gerard J. Murphy: C*-algebras and operator theory, Academic Press, 1990. You need to find yourself a copy of this book to follow the course.

Other literature

For some more advanced topics, we may also refer to the following book: Masamichi Takesaki: Theory of operator algebras I. Springer, 1979, 2001.

A useful book giving many non-trivial examples of C*-algebras is: Kenneth R. Davidson: C*-Algebras by example. American Mathematical Society, 1996.

Lecturer

M. Caspers (TUD)