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.
We prove that ample groupoids with
σ
\sigma
-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matui’s notion of Kakutani equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph
C
∗
C^*
-algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras
L
Z
(
E
2
)
L_{\mathbb {Z}}(E_2)
and
L
Z
(
E
2
−
)
L_{\mathbb {Z}}(E_{2-})
are not stably
∗
^*
-isomorphic.
We show that if A is a unital C * -algebra and B is a Cuntz-Krieger algebra for which A ⊗ K ∼ = B ⊗ K, then A is a Cuntz-Krieger algebra. Consequently, corners of Cuntz-Krieger algebras are Cuntz-Krieger algebras.
We prove that if E and F are graphs with a finite number of vertices and an infinite number of edges, if K is a field, and if LK (E) and LK (F ) are simple Leavitt path algebras, then LK (E) is Morita equivalent to LK (F ) if and only if K alg 0 (LK (E)) ∼ = K alg 0 (LK (F )) and the graphs E and F have the same number of singular vertices, and moreover, in this case one may transform the graph E into the graph F using basic moves that preserve the Morita equivalence class of the associated Leavitt path algebra. We also show that when K is a field with no free quotients, the condition that E and F have the same number of singular vertices may be replaced by K alg 1 (LK(E)) ∼ = K alg 1 (LK (F )), and we produce examples showing this cannot be done in general. We describe how we can combine our results with a classification result of Abrams, Louly, Pardo, and Smith to get a nearly complete classification of unital simple Leavitt path algebras -the only missing part is determining whether the "sign of the determinant condition" is necessary in the finite graph case. We also consider the Cuntz splice move on a graph and its effect on the associated Leavitt path algebra.
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.
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