Course info

The M1 course "Functional Analysis" is a required prerequisite. We further assume familiarity with basics of tensor products of vector spaces.

Operator algebras originate from the late 1930's when John von Neumann proposed a new framework in terms of "rings of operators" to study quantum mechanics and which we call "von Neumann algebras" nowadays. Soon after, Israel Gelfand developed the more general theory of C*-algebras and laid further connections to representation theory. The theory of C*-algebras and von Neumann algebras form a single area that we call operator algebras. The field saw a century of remarkable developments and is nowadays tightly connected to several mathematical areas such as: representation theory, geometric and combinatorial group theory, logic & descriptive set theory, ergodic theory, topology, differential geometry, knot theory, probability, quantum information theory and recently there have even been links to theoretical computer science.

In a nutshell 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. One can think of all bounded operators on a Hilbert space or the compact operators as first examples of operator algebras. A fundamental theorem , proved in the course "Functional Analysis", is that every commutative C*-algebra can be identified as complex valued continuous functions on a (locally) compact Hausdorff space. Von Neumann algebras can always be identified with L-infinity functions on a good measure space or as a probability space. 

Starting from these cornerstone theorems the theory develops in many directions of which we will see some in the course. In the first part will focus on von Neumann algebras, representation theory and tensor products of C*-algebras. In the second part of the course we will focus on K-theory: this can be viewed as a group of homotopy classes of non-commutative vector bundles (though no knowledge of differential geometry is required for this course) and it connects the theory of operator algebras to topology and geometry via Fredholm index theory.

Part I: Theory of operator algebras

Part II: K-theory

We will post weekly exercises on the Mastermath website and provide some sketches and hints for the solutions to help you. We encourage people to discuss with other students. The ELO also has a forum which can be used to discuss with other students but which will not be moderated by us in principle. Every week we will use the 1st lecture hour as an exercise session. Depending on the needs we discuss questions of the previous week or make an exercise together.

During the semester, there will be 3 hand-in exercises that will be graded and can count towards the final grade as explained below.

At the end of the course there will be a written exam which is open book. Allowed sources are the books by Murphy and Roerdam, and the lecture notes by Mesland. No other materials are allowed.

The final grade for the course is the exam grade rounded off to integers if the grade is strictly below 5.0. If your exam grade is a 5.0 or higher, then your final grade is the number max( 25% takehome exercises + 75% exam grade, exam grade) rounded off at the nearest integer.

The retake is a written open book exam (same rules as for the exam). If you take the retake your grade is the grade for the retake (so no bonus for the take home exercises).