Kirchberg $X$-algebras with real rank zero and intermediate cancellation

Abstract

A universal coefficient theorem is proved for $C^{\ast }$-algebras over an arbitrary finite $T_0$-space $X$ which have vanishing boundary maps. Under bootstrap assumptions, this leads to a complete classification of unital/stable real-rank-zero Kirchberg $X$-algebras with intermediate cancellation. Range results are obtained for (unital) purely infinite graph $C^{\ast }$-algebras with intermediate cancellation and Cuntz–Krieger algebras with intermediate cancellation. Permanence results for extensions of these classes follow.