Total Cuntz semigroup, Extension and Elliott Conjecture with Real rank zero

Abstract: In this paper, we exhibit two unital, separable, nuclear ${\rm C}^*$-algebras of stable rank one and real rank zero with the same ordered scaled total K-theory, but they are not isomorphic with each other, which forms a counterexample to Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of total K-theory. For the general setting, with a new invariant -- total Cuntz semigroup \cite{AL}, we classify a large class of ${\rm C}^*$-algebras obtained from extensions. The total Cuntz semigroup, which distinguish the algebras of our counterexample, could possibly classify all the ${\rm C}^*$-algebras of stable rank one and real rank zero.