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We associate to each row-finite directed graph E a universal Cuntz-Krieger C - algebra C (E), and study how the distribution of loops in E affects the structure of C (E). We prove that C (E) is AF if and only if E has no loops. We describe an exit condition (L) on loops in E which allows us to prove an analogue of the Cuntz-Krieger uniqueness theorem and give a characterisation of when C (E) is purely infinite. If the graph E satisfies (L) and is cofinal, then we have a dichotomy: if E has no loops, then C (E) is AF; if E has a loop, then C (E) is purely infinite.

Abstract. We prove versions of the fundamentaltheorems about Cuntz-Krieger algebras for the C ∗-algebras of row-finite graphs: directed graphs in which each vertex emits at most finitely many edges. Special cases of these results have previously been obtained using various powerful machines; our main point is that direct methods yield sharper results more easily. Contents 1. The C∗-algebras of graphs 309 2. The gauge-invariant uniqueness theorem 311 3. The Cuntz-Krieger uniqueness theorem 313 4. Ideals in graph algebras 316

Abstract. We classify the gauge-invariant ideals in the C ∗-algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant primitive ideals in terms of the structural properties of the graph, and describe the K-theory of the C ∗-algebras of arbitrary infinite graphs. 1.

Abstract. To an arbitrary directed graph we associate a row-finite directed graph whose C ∗-algebra contains the C ∗-algebra of the original graph as a full corner. This allows us to generalize results for C ∗-algebras of row-finite graphs to C ∗-algebras of arbitrary graphs: the uniqueness theorem, simplicity criteria, descriptions of the ideals and primitive ideal space, and conditions under which a graph algebra is AF and purely infinite. Our proofs require only standard Cuntz-Krieger techniques and do not rely on powerful constructs such as groupoids, Exel-Laca algebras, or Cuntz-Pimsner algebras. 1.

We construct a universal Cuntz-Krieger algebra AO A , which is isomorphic to the usual Cuntz-Krieger algebra O A when A satises the condition (I) of Cuntz and Krieger. Cuntz's classication of ideals in O A when A satises condition (II) extends to a classication of the gauge invariant ideals in AO A . We use this to describe the topology on the primitive ideal space of AO A . 1 Introduction In [4] Cuntz and Krieger studied C -algebras generated by families of n nonzero partial isometries S i satisfying S i S i = n X i=1 A(i; j)S j S j and n X i=1 S i S i = 1; (1) where A is an nn matrix with entries in f0; 1g and no zero rows or columns. It was shown in [4, Theorem 2.13] that, if A satises a certain condition (I), then C (S i ) is unique up to canonical isomorphism (i.e., an isomorphism mapping generators to generators), so that O A := C (S i ) depends only on A. The algebra O A is simple if A is irreducible and not a permutation matrix [4, ...

We give necessary andsufficient conditions for simplicity of Cuntz-Krieger algebras corresponding to infinite 0–1 matrices andof C ∗-algebras corresponding to countable directed graphs. We show that simple algebras within these two classes are either purely infinite or AF. 0. Since their invention about twenty years ago Cuntz-Krieger algebras OA corresponding to finite, square 0–1 matrices [3] have attracted immense interest. It was remarked in their original paper by Cuntz and Krieger that the theory may be extended to infinite matrices as well. Unfortunately, no details were provided at that time. This quite non-trivial task has been successfully carried out recently by Exel and Laca [5], by means of some heavy-duty machinery. It turns out that the resulting class of C ∗-algebras is rich enough to encompass, at least up to Morita equivalence, all graph C ∗-algebras [10, 9, 1, 11], as shown in [6], as well as AF-algebras, as shown

We dedicate this work to Yuri Manin, with admiration and gratitude In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the “closed fibers at infinity”. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus g ≥ 2. We use Connes ’ theory of spectral triples to relate the hyperbolic geometry of the handlebody to Deninger’s Archimedean cohomology and the cohomology of the cone of the local monodromy N at arithmetic infinity as introduced by the first author of this paper. First, we consider derived (cohomological) spectral data (A, H · (X ∗),Φ), where the algebra is obtained from the SL(2, R) action on the cohomology of the cone, induced by the presence of a polarized Lefschetz module structure, and its restriction to the group ring of a Fuchsian Schottky group. In this setting we recover the alternating product of the Archimedean factors from a zeta function of a spectral triple. Then, we introduce a different construction, which is related to Manin’s description of the dual graph of the fiber at infinity. We

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We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of one--sided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions their C --algebras are Morita equivalent; the groupoid C --algebra C (G) is the Cuntz--Krieger algebra of an infinite f0; 1g matrix defined by G, and that the algebras C (G(?)) contain the C --algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoid C --algebras using the general theory of Renault, and calculate their K-theory. 1 Introduction Over the past fifteen years many C -algebras and classes of C -algebras have been given groupoid models. Here we consider locally finite directed graphs, which may have infinitely many vertices, but only finitely many edges in and out of each vertex. We associate ...