Filtrated K-theory for real rank zero C*-algebras

Using Kirchberg KK_X-classification of purely infinite, separable, stable, nuclear C*-algebras with finite primitive ideal space, Bentmann showed that filtrated K-theory classifies purely infinite, separable, stable, nuclear C*-algebras that satisfy that all simple subquotients are in the bootstrap class and that the primitive ideal space is finite and of a certain type, referred to as accordion spaces. This result generalizes the results of Meyer-Nest involving finite linearly ordered spaces. Examples have been provided, for any finite non-accordion space, that isomorphic filtrated K-theory does not imply KK_X-equivalence for this class of C*-algebras. As a consequence, for any non-accordion space, filtrated K-theory is not a complete invariant for purely infinite, separable, stable, nuclear C*-algebrass that satisfy that all simple subquotients are in the bootstrap class. In this paper, we investigate the case for real rank zero C*-algebras and four-point primitive ideal spaces, as this is the smallest size of non-accordion spaces. Up to homeomorphism, there are ten different connected T_0-spaces with exactly four points. We show that filtrated K-theory classifies purely infinite, real rank zero, separable, stable, nuclear C*-algebras that satisfy that all simple subquotients are in the bootstrap class for eight out of ten of these spaces.