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Counterexamples to classification of purely infinite, nuclear, separableC*-algebras (in the ideal-related bootstrap class) and with primitive idealspace X using ideal-related K-theory occur for infinitely many finite primitiveideal spaces X, the smallest of which having four points. Ideal-relatedK-theory is known to be strongly complete for such C*-algebras if they havereal rank zero and X has at most four points for all but two exceptionalspaces: the pseudo-circle and the diamond space. In this article, we closethese two remaining cases. We show that ideal-related K-theory is stronglycomplete for real rank zero, purely infinite, nuclear, separable C*-algebrasthat have the pseudo-circle as primitive ideal space. In the oppositedirection, we construct a Cuntz-Krieger algebra with the diamond space as itsprimitive ideal space for which an automorphism on ideal-related K-theory doesnot lift.

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