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. I plan to stream and record the lectures in this class to allow some external participants to follow this course. Click here to go to the page where streams are listed. You need to pick the stream from Hoersaal Sitzungszimmer, which is where the course takes place.
A couple of classes will have to be shifted to other dates, which may or may not change the room. You may find the dates when classes are scheduled to take place in the table below, together with links to the recorded lectures, once these become available. Exercise sheets for this class are also listed below. Sample solutions may be provided upon request.
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. Then I will move on to study Hilbert modules over C*-algebras. These 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, except perhaps for a few lectures that focus on representation theory. I do assume knowledge of functional analysis, however. If you consider writing a bachelor thesis with me in this direction, then please mention this to me in April or May. It makes sense to start work on it during the semester.