Classification of real rank zero, purely infinite C⁎-algebras with at most four primitive ideals

Counterexamples to classification of purely infinite, nuclear, separable C⁎C⁎-algebras (in the ideal-related bootstrap class) and with primitive ideal space X using ideal-related K-theory occur for infinitely many finite primitive ideal spaces X  , the smallest of which being six spaces with four points. All constructed counterexamples for spaces with at least five points are based on the counterexamples constructed for these six four-point spaces. With real rank zero added to the assumptions imposed on the C⁎C⁎-algebras, ideal-related K-theory is known to be strongly complete for four of these six spaces. In this article, we close the two remaining cases: the pseudo-circle and the diamond space. We show that ideal-related K-theory is strongly complete for real rank zero, purely infinite, nuclear, separable C⁎C⁎-algebras that have the pseudo-circle as primitive ideal space. In the opposite direction, we construct a Cuntz–Krieger algebra with the diamond space as its primitive ideal space for which an automorphism on ideal-related K-theory does not lift.