24th Internetseminar
I am very happy to announce the 24th Internetseminar on "C*-algebras and dynamics" as well as to be a part of the organizing team. Below, you can find more information as it has been announced by us . If there are any questions please do not hesitate to ask.
ISem24 - C*-algebras and dynamics
From October 2020 until June 2021, the 24th Internet seminar (ISem24) will take place addressing Master’s and PhD students from all over the world interested in functional analysis. This year’s topic is „C*-algebras and dynamics“. If you are interested in receiving further information about ISem24, please inscribe to our mailing list by sending an email with your name and your affiliation to:
We are looking forward to welcoming you to our online event! Please forward this information to anyone who might be interested.
Organizers:
Christian Budde (North-West U, South Africa)
Moritz Weber (Saarbruecken, Germany)
Lecturers:
Xin Li (Glasgow, UK)
Christian Voigt (Glasgow, UK)
Moritz Weber (Saarbruecken, Germany)
Lecture Phase (October 2020 - February 2021):
Electronic lecture notes are provided weekly via the ISem24 website. 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 at the Bundeshoehe in Wuppertal, Germany. The projects from the Project Phase are presented and there are talks by experts in the field of C*-algebras.
Contents:
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 satisfying a particular norm condition. This condition forces a behavior similar to the supremum norm on the algebra C(X) of continuous, complex-valued functions on a compact Hausdorff space X. Such algebras are commutative C*-algebras and it is a fundamental theorem by Gelfand and Naimark that the converse is also true: For any (unital) commutative C*-algebra there exists a compact space 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 functional analysis and on (bounded linear operators on) Hilbert spaces, we will spend two thirds of the lecture for introducing C*-algebras, spectra of Banach algebras and proving 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. Questions such as simplicity, the implications of faithful/free/etc actions for the crossed products and further aspects will be discussed. We may thus use techniques from C*-algebras in order to learn something about classical dynamical systems.
About the Internet seminar:
The Internet Seminar (ISem) has been organized every year since 1997 by several research groups from Austria, Germany, Hungary, Italy and the Netherlands. It has been founded by the functional analysis groups in Tuebingen, Ulm and Karlsruhe (all in Germany). The ISem introduces Master's and PhD students to modern topics in functional analysis related to evolution equations. It is structured in three phases and with ISem24, we follow this structure. Usually, there are some hundreds of participants from all over the world.
Announcement as a PDF:
https://www.math.uni-sb.de/ag/speicher/weber/Isem24Announcement.pdf
Poster of ISem24:
https://www.math.uni-sb.de/ag/speicher/weber/ISem24Posterv2.pdf
Interim website of ISem24: