Nicolai Stammeier, Münster:From independence to *-commutativity and back

Abstract: Independence has been introduced as a regularity property for pairs of commuting injective group endomorphisms of a discrete abelian group with finite cokernel by Joachim Cuntz and Anatoly Vershik. We discuss various characterisations of this regularity property and show how the statements need to be adjusted when removing the restrictions that the group has to be abelian and that the cokernels have to be finite. Somewhat surprisingly, this leads to the concept of *-commutativity. This property is defined for pairs of commuting self-maps of an arbitrary set. As an examples of *-commutativity, we explain a construction related to the Ledrappier shift and indicate how one obtains examples for independent group endomorphisms from this construction. If time permits, we will point out instances where the two notions have been readily used to obtain C*-algebraic results. Roughly speaking, both notions are designed to give rise to pairs of doubly commuting isometries, which significantly simplifies the analysis of the constructed C*-algebras. This is particularly useful when one tries to generalise results from the case of a single transformation to an action generated by finitely many transformations.