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Mathematics Münster

B: Spaces and Operators

B3: Operator algebras & mathematical physics
Current ProjectsCRC 1442: Geometry: Deformation and Rigidity - D01: Amenable dynamics via C*-algebras Three regularity properties and their interplay have been at the heart of exciting recent developments in the structure and classification theory of nuclear C*-algebras: finite nuclear dimension, tensorial absorption of the Jiang–Su algebra, and strict comparison of positive elements. There are corresponding properties for group actions; we will study these dynamic regularity properties in order to gain new insights into amenable groups and their actions, and on rigidity properties of their associated C*-algebras. Taking a dual viewpoint, we will study - and to some extent classify - Cartan subalgebras of C*-algebras. These are maximal abelian subalgebras, the position of which encapsulates crucial information about the underlying dynamics of a C*-algebra. online
Amenability, Approximation and Reconstruction Algebras of operators on Hilbert spaces were originally introduced as the right framework for the mathematical description of quantum mechanics. In modern mathematics the scope has much broadened due to the highly versatile nature of operator algebras. They are particularly useful in the analysis of groups and their actions. Amenability is a finiteness property which occurs in many different contexts and which can be characterised in many different ways. We will analyse amenability in terms of approximation properties, in the frameworks of abstract C*-algebras, of topological dynamical systems, and of discrete groups. Such approximation properties will serve as bridging devices between these setups, and they will be used to systematically recover geometric information about the underlying structures. When passing from groups, and more generally from dynamical systems, to operator algebras, one loses information, but one gains new tools to isolate and analyse pertinent properties of the underlying structure. We will mostly be interested in the topological setting, and in the associated C*-algebras. Amenability of groups or of dynamical systems then translates into the completely positive approximation property. Systems of completely positive approximations store all the essential data about a C*-algebra, and sometimes one can arrange the systems so that one can directly read of such information. For transformation group C*-algebras, one can achieve this by using approximation properties of the underlying dynamics. To some extent one can even go back, and extract dynamical approximation properties from completely positive approximations of the C*-algebra. This interplay between approximation properties in topological dynamics and in noncommutative topology carries a surprisingly rich structure. It connects directly to the heart of the classification problem for nuclear C*-algebras on the one hand, and to central open questions on amenable dynamics on the other. online
EXC 2044 - B3: Operator algebras & mathematical physics The development of operator algebras was largely motivated by physics since they provide the right mathematical framework for quantum mechanics. Since then, operator algebras have turned into a subject of their own. We will pursue the many fascinating connections to (functional) analysis, algebra, topology, group theory and logic, and eventually connect back to mathematical physics via random matrices and non-commutative geometry. online
Phone+49 251 83-33756
Secretary   Elke Enning
Frau Elke Enning
Telefon +49 251 83-33088
Fax +49 251 83-32708
Zimmer 402
AddressProf. Dr. Wilhelm Winter
Mathematisches Institut
Fachbereich Mathematik und Informatik der Universität Münster
Einsteinstrasse 62
48149 Münster
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