On the Independence of K-Theory and Stable Rank for Simple C*-Algebras

Abstract

Jiang and Su and (independently) Elliott discovered a simple, nuclear, infinite-dimensional C*-algebra ℒ̵ 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 ⊗ ℒ̵ is isomorphic to that of A. Thus, any simple nuclear C*-algebra A having a weakly unperforated K*-group which does not absorb ℒ̵ provides a counterexample to Elliott's conjecture that the simple nuclear C*-algebras will be classified by the Elliott invariant. In the sequel we exhibit a separable, infinite-dimensional, stably finite instance of such a non-ℒ̵-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.