Logic Seminar

SEMINAR WILL BE HELD IN 159 SLOAN

There exist examples of compact Hausdorff spaces with no non-trivial homeomorphisms, and in particular with no non-trivial group actions on them. In the opposite extreme, there are many spaces that have a huge number of homeomorphisms. Since C*-algebras are noncommutative generalizations of spaces, we may ask whether a given (separable) C*-algebra has `many' automorphisms, or, more generally, `many' actions of a given group. In this context, we must of course identify actions up to a reasonable notion of equivalence, since otherwise most C*-algebras will have uncountably many actions. We are not only interested in knowing ``how many" actions there are, but also determining the Borel complexity of the relation of equivalence between them. Questions of this sort have been studied in the context of von Neumann algebras, and particularly the hyperfinite II$_1$-factor $\mathcal{R}$. In this setting, the combination of a famous theorem of Ocneanu with a recent result by Brothier-Vaes shows that there is a dichotomy for the cardinality of the set of outer actions of a given group on $\mathcal{R}$: for an amenable group, there is a unique one, while for nonamenable groups there are uncountably many. In the context of C*-algebras, one may want to replace the hyperfinite II$_1$-factor with a UHF-algebra of infinite type. In this setting, far less is known, and all the uniqueness results that are available so far work only for abelian groups (but not even all of them). In this talk, I will report on some recent joint work with Martino Lupini, where we establish the existence of uncountably many, non equivalent ``free" actions of a given group with property (T) on a UHF-algebra. In fact, we show that the relation of equivalence for these actions is a complete analytic set.