Operator algebra seminar

Speaker:  Matias Andersen (KU)

Title: Why Tarski’s theorem fails in a topological setting

Abstract: Given an action of a group on a set, a subset is called paradoxical in case it can be split up into a finite number of pieces that, via the action, can be assembled to two disjoint copies of the original subset and possibly some leftovers. If a subset is normalised by a finitely additive invariant measure, clearly it cannot be paradoxical, and Tarski’s theorem states that this is in fact the only obstruction.

However, if one is considering actions on totally disconnected topological spaces, it is reasonable to allow only decompositions that respect the topology, i.e. decompositions into clopen subspaces. In this talk, we will consider a class of partial actions introduced by Ara and Exel and indicate why they lead to a failure of Tarski’s theorem in this topological setting. If time permits, I will also discuss possible implications for C*-algebras.