Please visit the webpage of the lectures

The current state of the lecture notes is also available on this webpage, see the quick link on the left hand side.


Christian Budde (North-West U, South Africa) personal webpage
Moritz Weber (Saarbrücken, Germany) personal webpage


Xin Li (Glasgow, UK) personal webpage
Christian Voigt (Glasgow, UK) personal webpage
Moritz Weber (Saarbrücken, Germany) personal webpage


The registration deadline has passed.

Lecture Phase

October 2020 - February 2021
Electronic lecture notes are provided weekly via the ISem24 website starting from mid October. The lectures are self-contained and they include some exercises. In local groups, ideally led by a local coordinator, students from all over the world read these notes and discuss them in an online chat room.

Project Phase

March 2021 - June 2021
In small international groups led by some of the coordinators, the participants work on various projects supplementing the Lecture Phase.

Final Workshop

6-12 June 2021
A one-week workshop takes place virtually via Zoom. The projects from the Project Phase are presented and there are talks by experts in the field of C*-algebras.


In the 1940s, Gelfand and Naimark introduced C*-algebras, mainly in order to study representations of groups. It quickly developed into a research area on its own linking techniques from functional analysis and algebra in a fascinating way.
Technically speaking, C*-algebras are Banach algebras which are equipped with an involution satifying a particular norm condition:

                  ||x* x||=||x||2

This condition forces a behaviour similar to the supremum norm on the algebra C(X) of continuous, complex-valued functions on a compact Hausdorff space X. Indeed, such algebras are prototypes of commutative C*-algebras and it is a fundamental theorem by Gelfand and Naimark that the converse is also true: For any commutative C*-algebra there exists a compact space X such that the C*-algebra is isomorphic to C(X).
This is the famous Gelfand duality between compact topological spaces and commutative C*-algebras - turning the theory of (possibly noncommutative) C*-algebras into a kind of noncommutative topology.

Noncommutative C*-algebras come into play as soon as we move to dynamical systems, i.e. to actions of compact groups G on compact spaces X. Such dynamics may be studied in terms of the (typically noncommutative) C*-algebra given by the crossed product of C(X) with G and we may employ tools from the theory of C*-algebras.

Building on some basic knowledge on operators on Hilbert spaces, we will spend two thirds of the lecture to introduce C*-algebras, spectra of Banach algebras and prove Gelfand duality, which provides us with the powerful tool of functional calculus for continuous functions. We will then turn to unitizations, positive elements, approximate units, ideals, states and representations, eventually proving that all C*-algebras may be represented concretely on a Hilbert space.

In the last third of the lecture, we will study actions of groups on topological spaces, dynamical systems and crossed products of C*-algebras.

If you want to get an impression about the contents of ISem24, take a look at the slides of the talk given by Moritz Weber at the ISem23 Workshop in June 2020.


We expect the participants to have some basic knowledge in functional analysis, in particular regarding Hilbert spaces.

Ideally, the participants have some additional knowledge on the following subjects. However, let us stress that this is not a strict requirement - we will introduce/recall all the relevant facts from the list below that are needed throughout the lectures.
These topics are for instance covered in the following lecture notes by Moritz Weber. See also here for some more advanced lecture notes covering parts of the contents of ISem24.

About the Internet Seminar series

The Internet Seminar (ISem) has been organized every year since 1997 by several working groups from Austria, Germany, Hungary, Italy and the Netherlands. It has been founded by the functional analysis groups in Tübingen, Ulm and Karlsruhe. The ISem introduces Master's and PhD students to modern topics in functional analysis related to evolution equations. See here for the history of the ISem and its structure in three phases. With ISem24, we follow this structure. Usually, there are some hundreds of participants from all over the world.

Some advertising: international foundation programme VSi MINT

The international foundation programme VSi MINT has been designed to enable candidates who do not have a recognized higher education entrance qualification and whose German is not yet proficient enough for normal study to join Saarland University's special Bachelor Plus MINT programme.
(Note: The German acronym 'MINT' is equivalent to the English acronym 'STEM', which stands for Science, Technology, Engineering, Mathematics.)

Last update: 30 March 2021   Moritz Weber Impressum