Sven Raum: C*-superrigidity

Abstract:
It is a classical problem to recover a discrete group from various rings or algebras associated with it, such as the integral group ring (cf. the Whitehead group and the Whitehead torsion). By analogy, in an operator algebraic framework we want to recover torsion-free groups from certain topological completions of the complex group ring, such as the reduced group C*-algebra. Groups for which this is possible are called C*-superrigid.

In this talk, I will introduce the topic of C*-superrigidity assuming only a background in basic group theory and group rings. The reduced group C*-algebras is put in the context of the more familiar complex group algebra. An important question on this notion asks whether all torsion-free discrete groups are C*-superrigid, paralleling Higman's unit conjecture for group rings. However, until 2017 the only known such examples of torsion-free C*-superrigid groups were abelian. To guarantee general understanding, it is exactly around the examples of abelian torsion-free groups that most of the talk will turn. Having motivated and described C*-superrigidity as a subject, I will give a brief account of the developments since 2017 including my own contribution to the subject. The talk will finish with a perspective on possible impact of C*-superrigidity in other areas of mathematics.