On the independence of K-theory and stable rank for simple C-algebras

Jiang and Su and (independently) Elliott discovered a simple, nuclear, infinite-dimensional C-algebra Z having the same Elliott invariant as the complex numbers. For a nuclear C-algebra A with weakly unperforated K∗-group the Elliott invariant of A ⊗ Z is isomorphic to that of A. Thus, any simple nuclear C-algebra A having a weakly unperforated K∗-group which does not absorb Z provides a counterexample to Elliott’s conjecture that the simple nuclear Calgebras will be classified by the Elliott invariant. In the sequel we exhibit a separable, infinite-dimensional, stably finite instance of such a non-Z-absorbing algebra A, and so provide a counterexample to the Elliott conjecture for the class of simple, nuclear, infinite-dimensional, stably finite, separable C-algebras.