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

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 having four points. Ideal-related K-theory is known to be strongly complete for such C*-algebras if they have real rank zero and X has at most four points for all but two exceptional spaces: the pseudo-circle and the diamond space. In this article, we close these two remaining cases. 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.