On the classification problem for nuclear $C^\ast$-algebras

Abstract

We exhibit a counterexample to Elliott’s classification conjecture for simple, separable, and nuclear C$^*$-algebras whose construction is elementary, and demonstrate the necessity of extremely fine invariants in distinguishing both approximate unitary equivalence classes of automorphisms of such algebras and isomorphism classes of the algebras themselves. The consequences for the program to classify nuclear C$^*$-algebras are far-reaching: one has, among other things, that existing results on the classification of simple, unital AH algebras via the Elliott invariant of $\mathrm{K}$-theoretic data are the best possible, and that these cannot be improved by the addition of continuous homotopy invariant functors to the Elliott invariant.