Total Cuntz semigroup, extension, and Elliott Conjecture with real rank zero

In this paper, we exhibit two unital, separable, nuclear ${\mathrm{C}}^{\ast }$-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 the Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of the total K-theory. For the general setting, with a new invariant, the total Cuntz semigroup [2], we classify a large class of ${\mathrm{C}}^{\ast }$-algebras obtained from extensions. The total Cuntz semigroup, which distinguishes the algebras of our counterexample, could possibly classify all the ${\mathrm{C}}^{\ast }$-algebras of stable rank one and real rank zero.