An introductory course in functional analysis is required. You must be familiar with notions as Hilbert spaces, projections and closed subspaces, bounded operators on a Hilbert space, the Hahn-Banach theorem and preferably its consequences for the separation of sets with functionals, Banach-Alaoglu theorem, spectrum of operators (but not necessarily the spectral theorem), compact operators, basic examples of Banach spaces such as Lp-spaces (mostly for p=2 or p=infinity).
Aim of the course
Operator algebras originate from the work of Johan von Neumann and Israel Gelfand who had the aim to understand both quantum mechanics and representation theory. The theory grew into an entire mathematical discipline with several subfields and important connections to knot theory, probability theory, group theory and dynamical systems.
Operator algebras concerns the theory of algebras of bounded operators on a Hilbert space. Thus instead of studying a single operator one rather studies algebras of operators as a whole. Essentially the theory splits into two parts: C*-algebras and von Neumann algebras. A commutative C*-algebra can always be identified as continuous functions on a compact Hausdorff space (Gelfand-Naimark theorem) and as such C*-algebras should be viewed as a non-commutative generalisation of the notion of topological space. This is one of the main theorems at the start of this course. Similarly, commutative von Neumann algebras can be identified with a measure space, and therefore play an important role in quantum probability.
The aim of the first half of this course is to give an introduction to the theory of C*-algebras and von Neumann algebras. In the second part of the course we will look at one of the more advanced tools that are used in the classification of C*-algebras: K-theory, which are non-commutative analogues of vector bundles.
The precise topics treated in this course are:
-Gelfand Naimark theorem characterising commutative C*-algebras.
-Positive cones in C*-algebras.
-States on C*-algebras and the GNS-construction.
-Unitizations of C*-algebras.
-von Neumann's double commutant theorem for von Neumann algebras.
-Characterisations of von Neumann algebra's in terms of the strong and weak operator topology. Predual of a von Neumann algebra.
-Trace class and Hilbert-Schmidt operators.
-Characterisation of commutative von Neumann algebras in terms of measure spaces.
-Kaplansky's density theorem.
-Definition and first properties of K-theory and computation of the K-theory of B(H) and K(H).
-AF-algebras and the Elliott classification in terms of K-theory.
Martijn Caspers (TU Delft) for the first part
Bram Mesland (Universiteit Leiden) for the second part