A Class of C∗-algebras Generalizing Both Graph Algebras and Homeomorphism C∗-algebras Iv, Pure Infiniteness

This is the final one in the series of papers where we introduce and study the C∗-algebras associated with topological graphs. In this paper, we get a sufficient condition on topological graphs so that the associated C∗-algebras are simple and purely infinite. Using this result, we give one method to construct all Kirchberg algebras as C∗-algebras associated with topological graphs. 0. Introduction The classification theory of simple separable nuclear C∗-algebras by K-theory has been rapidly developed recently. This classification was completed for the purely infinite case independently by Kirchberg [Ki] and Phillips [P]. Recall that a simple C∗-algebra is said to be purely infinite if every non-zero hereditary C∗-subalgebra has an infinite projection. Definition. A Kirchberg algebra is a simple, separable, nuclear, purely infinite C∗-algebra satisfying the universal coefficient theorem of [RS]. Theorem (Kirchberg, Phillips). Two non-unital Kirchberg algebras A and B are isomorphic if and only if (K0(A), K1(A)) ∼= (K0(B), K1(B)), and two unital Kirchberg algebras A and B are isomorphic if and only if (K0(A), [1A], K1(A)) ∼= (K0(B), [1B], K1(B)). See Rørdam’s book [Rø] for detailed definitions and a proof of the above theorem (note that in [Rø] a Kirchberg algebra is not assumed to satisfy the universal coefficient theorem). There are many ways to construct Kirchberg algebras, and all pairs (G0, G1) of countable abelian groups appear as K-groups of non-unital Kirchberg algebras, and for all g ∈ G0 there exists a unital Kirchberg algebra A with (K0(A), [1A], K1(A)) ∼= (G0, g, G1) (see [Rø, Subsection 4.3]). One way to construct Kirchberg algebras is by graph algebras. A graph algebra is a C∗-algebra associated with a (directed) graph, which generalizes Cuntz-Krieger algebras defined in [CK] (see [Ra] for a definition and properties of graph algebras). We know a necessary and sufficient condition on graphs so that the associated graph algebras are simple and purely infinite, and in this case the graph algebras become Kirchberg algebras (see [Ra, Remark 4.3]). However we cannot construct all Kirchberg algebras by graph algebras because K1-groups of graph algebras are always free. Spielberg constructed all non-unital Kirchberg algebras mixing the constructions of graph algebras and the higher rank graph algebras [Sp1]. In this paper, we examine for which topological graphs the associated C∗-algebras are simple and purely infinite. We also construct all Kirchberg algebras as C∗-algebras associated with topological graphs. A topological graph is a quadruple E = (E, E, d, r) 2000 Mathematics Subject Classification. Primary 46L05; Secondary 46L55, 37B99.