C*-Algebras, towards Cuntz-Pimsner algebras of C*-correspondences (Prof. Dr. Ralf Meyer)

Welcome to my class on C*-algebras in the summer term 2024! The class starts with a general introduction to C*-algebras and then moves on to cover Hilbert modules and C*-correspondences and their Toeplitz and Cuntz-Pimsner algebras.

Classes are scheduled on Tuesday and Friday 8-10, from April, 9th, 2024 until July, 12th, 2024, in the Hörsaal Sitzungszimmer in the mathematical institute. The lectures in this class were streamed and recorded to allow some external participants to follow this course. The recordings are listed below.

Course outline

This course begins with the general theory of C*-algebras, up to the Gelfand-Naimark Theorem, which realises any C*-algebra as a C*-subalgebra of bounded operators on a Hilbert space. This is roughly half of the course, and fairly standard. In the second half, I will move on to study Hilbert modules over C*-algebras and their Toeplitz and Cuntz-Pimsner C*-algebras. Hilbert modules generalise Hilbert spaces by allowing a module over a C*-algebra instead of a vector space, equipped with a C*-algebra valued inner product. Such a Hilbert module over a C*-algebra together with a representation of the C*-algebra on it is called a C*-correspondence. This is the initial data for the construction of Cuntz-Pimsner algebras. This is an important method to define interesting C*-algebras. In particular, graph C*-algebras or C*-algebras of self-similar groups are defined in this way.

Cuntz-Pimsner algebras also figure prominently in my own research. One important aspect in my research is that C*-correspondences form a bicategory and that bicategory theory offers a useful perspective on constructions of C*-algebras such as Cuntz-Pimsner algebras. This would be a good direction for Bachelor and Master thesis under my direction. I plan, however, to focus on the analytical aspects of the Cuntz-Pimsner algebra construction, leaving the bicategorical links to individual reading or a separate class, which may be offered depending on demand and capacity.

Group representations may be studied using group C*-algebras and crossed products by group actions on C*-algebras. This links this course to the Harmonic Analysis course in the previous term. Nevertheless, students who missed the Harmonic Analysis course may still do fine in this class. I do assume knowledge of functional analysis, however.

Lectures and recording links

Exercise sheets

• Exercise sheet 1
• Exercise sheet 2
• Exercise sheet 3
• Exercise sheet 4
• Exercise sheet 5
• Exercise sheet 6
• Exercise sheet 7
• Exercise sheet 8
• Exercise sheet 9
• Exercise sheet 10
• Exercise sheet 11
• Exercise sheet 12
• Exercise sheet 13

Lecture notes

• Hilbert modules