Abstract. We address the classi cation problem for graph C * -algebras of nite graphs ( nitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition (K), so that the graph C * -algebras may come with uncountably many ideals.We nd that in this generality, stable isomorphism of graph C * -algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the C * -algebras having isomorphic K-theories. is proves in turn that under this condition, the graph C * -algebras are in fact classi able by K-theory, providing in particular complete classi cation when the C * -algebras in question are either of real rank zero or type I/postliminal. e key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs.Our results are applied to discuss the classi cation problem for the quantum lens spaces de ned by Hong and Szymański, and to complete the classi cation of graph C * -algebras associated to all simple graphs with four vertices or less.

Inspired by Franks' classification of irreducible shifts of finite type, we provide a short list of allowed moves on graphs that preserve the stable isomorphism class of the associated C * -algebras. We show that if two graphs have stably isomorphic and simple unital algebras then we can use these moves to transform one into the other.

We prove that every Kirchberg algebra in the UCT class has nuclear dimension 1. We first show that Kirchberg 2-graph algebras with trivial K 0 and finite K 1 have nuclear dimension 1 by adapting a technique developed by Winter and Zacharias for Cuntz algebras. We then prove that every Kirchberg algebra in the UCT class is a direct limit of 2-graph algebras to obtain our main theorem.

Given a compact metric space X, we show that the commutative C * -algebra C(X) is semiprojective if and only if X is an absolute neighbourhood retract of dimension at most 1. This confirms a conjecture of Blackadar.Generalizing to the non-unital setting, we derive a characterization of semiprojectivity for separable, commutative C * -algebras. As applications of our results, we prove two theorems about the structure of semiprojective commutative C * -algebras. Letting A be a commutative C * -algebra, we show firstly: If I is an ideal of A and A/I is finite-dimensional, then A is semiprojective if and only if I is; and secondly: A is semiprojective if and only if M2(A) is. This answers two questions about semiprojective C * -algebras in the commutative case.