Operator algebras attached to groups and group-like objects

Originally, one of the main motivations to introduce operator algebras was to study unitary representations of groups. And indeed, this idea turned out to be very fruitful and has led to interesting new developments as well as important open problems.

The idea of this PhD project is to study C*-algebras, a particular class of operator algebras, attached to groups, semigroups, and groupoids.

The construction of semigroup C*-algebras is of current interest. Recently, it became clear that K-theory for semigroup C*-algebras is related to the Baum-Connes conjecture, which is an important open problem in topological K-theory. Moreover, in another direction, it has become clear that C*-algebras of groupoids provide a bridge between geometric group theory, topological dynamics, and C*-algebras. The relevant notion on the C*-algebra side is the notion of Cartan subalgebras, which is a largely undeveloped area.

The goals of this PhD project is to explore the link between K-theory of semigroup C*-algebras and the Baum-Connes conjecture, and to study homological invariants of groupoids and Cartan subalgebras. This research project has the potential of shedding some light on long-standing open problems. At the same time, it leads to many interesting and feasible research problems.