Pure Maths Colloquium - Noncommutative covering dimension for C*-algebras and dynamical systems - Dr Joachim Zacharias (Glasgow) Seminar
- Time:
- 15:00 - 16:00
- Date:
- 12 May 2017
- Venue:
- Room 10037, Lecture Theatre 10B, Building 54, Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ
For more information regarding this seminar, please telephone Dr Jan Spakula on 023 8059 3137 or email Jan.Spakula@southampton.ac.uk .
Event details
Abstract: Various noncommutative generalisations of dimension have been considered and studied in the past decades. One more recent such concept is nuclear dimension. Its basic idea is to mimic covering dimension of topological spaces by regarding open covers as approximations of spaces. This dimension concept has turned out to be important in the classification programme of simple nuclear C*-algebras. Subsequently, Rokhlin dimension, a dimension concept for dynamical systems, has been introduced, first for Z-actions and more recently for actions of residually finite groups. In case of actions on spaces it corresponds to a kind of equivariant covering dimension. Actions of groups with finite asymptotic dimension and finite Rokhlin dimension, a property which is often automatic, are well-behaved with respect to nuclear dimension. Finally there are links between the asymptotic dimension in coarse geometry and nuclear dimension. In recent work of Guentner, Willett and Yu these connections could be extended to group actions on spaces and lined to Rokhlin dimension. We will give an introduction to these concepts and survey some applications and connections between them. This survey is based on work in collaboration with Hirshberg, Szabo, Winter, Wu and further work by numerous authors.
Speaker information
Dr Joachim Zacharias, University of Glasgow. My research interests are in C*-algebras and their applications. C*-algebras and their measure theoretical counterpart, von Neumann algebras, are algebras of operators on a Hilbert space. They were first motivated by quantum theory and occurred in group representations and ergodic theory. They can also be characterised abstractly as Banach algebras by a very natural norm condition. The understanding of C*-algebras and von Neumann algebras has seen remarkable progress in the past decades which has lead to far reaching classification results using methods originally developed in topology, notably K-theory. C*-algebras have occurred in many other branches of mathematics. C*-algebra theory is a rapidly growing field with many exiting open problems and many possible PhD projects. I am particularly interested in classification of simple nuclear C*-algebras, non commutative dimension concepts, dynamical systems, special examples of C*-algebras, in particular the very rich class of Cuntz algebras and various generalisations (graph, Pimsner, higher rank, continuous etc.), K-theory for those C*-algebras, approximation properties of C*- and von Neumann algebras, dynamical systems and their applications to C*-algebras, noncommutative geometry (spectral triples).