Classification of Nuclear, Simple C*-algebras

Abstract

The possibility that nuclear (or amenable) C*-algebras should be classified up to isomorphism by their K-theory and related invariants was raised in an article by Elliott [48] (written in 1989) in which he showed that a certain class of inductive limit algebras (AT-algebras of real rank zero) admits such a classification. Elliott made the inspired suggestion that his classification theorem perhaps covers all separable, nuclear C*-algebras of real rank zero, stable rank one, and with torsion free K 0 and K 1-groups. This was the first formulation of the Elliott conjecture. Evidence in favor of the conjection was shortly after provided by Elliott and Evans ([51]) who showed that all irrational rotation C*-algebras belong to the class covered by Elliott’s classification theorem. The Elliott conjecture was later modified so as to encompass all nuclear, separable, simple C*-algebras (not necessarily of stable rank one and real rank zero, and without the restrictions on the K-theory). Ed Effros writes about Elliott’s conjecture: “This was regarded as ridiculous by many (including myself), and we waited for the counter-examples to appear. We are still waiting.”