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

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Counterexamples to classification of purely infinite, nuclear, separable 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⁎-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⁎-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.