The K-theoretical range of Cuntz–Krieger algebras

Abstract

Abstract. We augment Restorff's classification of purely infinite Cuntz-Krieger algebras by describing the range of his invariant on purely infinite CuntzKrieger algebras. We also describe its range on purely infinite graph C * -algebras with finitely many ideals, and provide 'unital' range results for purely infinite Cuntz-Krieger algebras and unital purely infinite graph C * -algebras.

Reduction of filtered K-theory and a characterization of Cuntz-Krieger algebras

“…In Section 6, reduced filtered K-theory FK R is defined, and it is shown in Section 7 that the concrete filtered K-theory FK ST (A) of a real-rank-zero C * -algebra A over an EBP space satisfying that all subquotients have free K 1 -groups can be recovered from the reduced filtered K-theory FK R (A), see Corollary 7.15. This is of particular interest because of the range results from [3] for (unital) reduced filtered K-theory on (unital) purely infinite graph C * -algebras, see Theorem 6.12 (and 8.10). In order to proceed from reduced to concrete filtered K-theory in Section 7, an "intermediate" invariant is introducted, which serves only technical purposes.…”

“…In the companion paper [3], we determine the range of reduced filtered K-theory with respect to purely infinite Cuntz-Krieger algebras and graph C * -algebras.…”

The nuclear dimension of graphC⁎-algebras

“…Proof. By [1,Corollary 7.16] (see also [2]), there exists a row-finite directed graph E such that C * (E) ⊗ K ∼ = A ⊗ K. Now [35, Corollary 2.8(i)] and Theorem 4.1 imply that the nuclear dimension of A is 1.…”

“…Suppose that K 1 (A(x)) is free and that A(x) belongs to the Rosenberg-Schochet bootstrap category N for each x ∈ X. Recent results of Arklint, Bentmann, and Katsura ( [1] and [2]), show that A is stably isomorphic to C * (E) for some row-finite graph E. Since nuclear dimension is preserved by stabilization, these results imply that A has nuclear dimension 1. Evidence suggests that, more generally, if A is a separable, nuclear, purely infinite, tight C * -algebra over any finite topological space X and if K 1 (A(x)) is free and A(x) is in N for all x ∈ X, then A is a purely infinite graph C * -algebra.…”

“…In Section 3, again following [35,Section 7], we construct homomorphisms j m : C * (E(m)) → C * (E) ⊗ K and prove that if C * (E) has finitely many ideals, then the homomorphisms j m •ι m : C * (E) → C * (E) ⊗ K induce multiplication by m in the K-groups of every ideal and quotient of C * (E). In Section 4, we combine this with heavy machinery of [17,24,25] and a technique developed by Enders [12] to prove that C * (E) has nuclear dimension 1 when it is purely infinite and has finitely many ideals; we deduce that strongly purely infinite nuclear UCT C * -algebras whose primitive ideal spaces are finite accordion spaces have nuclear dimension 1 using the classification results of [2,1]. In a short section 5, we use our main result and a technique of Jeong-Park [16] to see that purely infinite graph C * -algebras with infinitely many ideals have nuclear dimension at most 2.…”