Johannes Christensen is awarded a Villum Young Investigator grant
With this funding, Johannes will explore innovative approaches to decomposing C*-algebras.
The theory of C*-algebras was partly introduced to construct a mathematical formalism for quantum statistical mechanics, and it traces its origins to the pioneering work of John von Neumann in the 1930s. It has now developed into an active research area with close connections to many other fields of mathematics, yet the problem of describing the ideal structure of C*-algebras still remains open. As is often the case in mathematics, an efficient strategy for understanding an object is to decompose it into simpler parts and then describe these simpler parts. In the theory of C*-algebras, this decomposition is obtained by describing the ideal structure of the C*-algebra. Even though this ideal structure is known for specific examples, the structure is in general still an enigma, even for many important classes of C*-algebras that connect the theory to other fields of mathematics, like dynamical systems.
In recent years, so-called groupoid C*-algebras have come to play a prominent role in C*-algebra theory, as it has become evident that they can model an enormous class of C*-algebras. The underlying groupoids have a lot of symmetries built into them, in the sense that they often contain an abundance of groups.
“The project aims at describing the ideal structure for a large class of groupoid C*-algebras by developing new methods for translating ideal information back and forth from a groupoid C*-algebra to the C*-algebras of the groups in the underlying groupoid. Together with an international collaborator, I have recently taken the first steps towards developing a framework that makes such translations possible. It has been known for decades that one can induce ideals from the underlying group C*-algebras to the groupoid C*-algebras, but my project will develop tools to better track which ideals can be induced and also techniques for reducing ideals from the groupoid C*-algebra to the underlying groups. This will make it possible to describe the ideal structure for many new classes of C*-algebras, and in the process, we will deepen our understanding of the objects modelling equilibrium in quantum statistical mechanics.”
Johannes explains
The Villum Young Investigator grant will fund two postdocs and a PhD student.