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.Comment: 17 page
