Recasting the Elliott conjecture

Abstract

Let A be a simple, unital, finite, and exact C*-algebra which absorbs the Jiang–Su algebra \({\mathcal{Z}}\) tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases—\({\mathcal{Z}}\) -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of \({\mathcal{Z}}\) -stable C*-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that \({\mathrm{K}}\) -theoretic invariants will classify separable and nuclear C*-algebras—with the recent appearance of counterexamples to its strongest concrete form.