Dynamical Systems Seminar

We give a general introduction to C*-algebras with a particular emphasis on those arising from étale groupoids. These include C*-algebras arising from topological dynamical systems and various other combinatorial objects like directed graphs. Indeed, étale groupoids provide a natural lens through which to examine various combinatorial/dynamical properties and their C*-algebraic counterparts (e.g. minimality vs simplicity). Time permitting we will also outline our recent noncommutative extension of the classic Gelfand duality which unifies classic work of Dauns and Hofmann on C*-bundles with the more recent Weyl groupoid construction of Kumjian and Renault.

Following Hochman, say that a countable group G has the strong topological Rohlin property (STRP) if it admits a continuous action on the Cantor space whose conjugacy class is comeager (in an appropriate topology). It is a well-known result, originally proved by Kechris and Rosendal and then re-proved by different methods by Akin, Glasner and Weiss, that there is a generic homeomorphism on the Cantor space - or in other words, that the group of integers has the STRP. It was later proved by Hochman that Z^d, for d>1, does not have the STRP, and by Kwiatkowska that free groups on finitely many generators do have the STRP. We show that a countable group G has the STRP if and only if for any finite set A with at least two elements, in the space of all closed subshifts of A^G equipped with the Vietoris topology the isolated points are dense. The latter can be more conveniently restated in the language of SFTs, which will be done during the talk. Some applications will be presented, including a new proof of the STRP for Z.