Speaker: George Elliott (Univ. Toronto)
Title: The classification of well behaved simple C*-algebras
Time/Date: 4:45-6:15pm, Thursday, July 25, 2019.
Abstract: The only question now is exactly what "well behaved" means. It should certainly include "separable" and "amenable" (= "nuclear"). It should certainly also include the UCT, although Lin very recently has announced a proof that this is automatic (for separable amenable C*-algebras). If one is to use the simplest invariant, namely, in the stable case, the K-groups together with the tracial cone (and the natural pairing) ---which (when the intrinsic preorder on K-theory is not mentioned) is invariant under tensoring with the Jiang-Su C*-algebra---then one must assume stability under tensoring with this algebra. In particular, this algebra has this property, so any simple separable amenable C*-algebra tensored with it is Jiang-Su stable---and in fact classifiable with this invariant! (There are also lots of examples for which Jiang-Su stability is known to hold naturally.) If one does not assume this property, then at the very least one will need an additional invariant. The Cuntz semigroup, since it has enough information (in what may be the general case) to determine whether an algebra is Jiang-Su stable, would appear to be a plausible candidate. (While I may not say much about it, the question of non-simple C*-algebras, even well behaved, also beckons!)