Prof. Dr. Moritz Weber

24th Internetseminar (ISem24)
on C*-algebras and dynamics

(Winter term 2020/2021)


See the official ISem24 page.

Time and Place

Thursday, 14-16m n.V. IV, building E2 5


This seminar is the local edition of ISem24, a series of internet seminars on functional analysis.
The lectures notes will be provided on the ISem24 webpage and we will read them in our local seminar.

The contents of ISem24 is as follows:

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.
  • Banach spaces, Hilbert spaces, dual spaces

  • Hahn-Banach Theorem

  • Krein-Milman Theorem

  • some knowledge on bounded linear operators on Hilbert spaces such as compact operators, spectral values, possibly even the Spectral Theorem for compact and/or normal operators

  • Spectra of Banach algebras

  • Stone-Weierstrass Theorem

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.

Last update: 29 May 2019   Moritz Weber Impressum