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.

In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twisted K-theory. We provide some examples. Possible applications of bundle gerbe K-theory to the classification of D-brane charges in nontrivial backgrounds are briefly discussed.

1 Transverse measure for flows 4 2 Transverse measure for foliations 6

Mathematicians tend to think of the notion of symmetry as being virtually synonymous with the theory of groups and their actions, perhaps largely because of the well-known Erlanger program

In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an S 1-gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framework for studying various twisted K-theories including the usual twisted K-theory of topological spaces, twisted equivariant K-theory, and the twisted K-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted K-groups can be expressed by so-called “twisted vector bundles”. Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of K-theory (KK-theory) of C ∗-algebras.

Abstract. Given a group G, we construct, in a canonical way, an inverse semigroup S(G) associated to G. The actions of S(G) areshowntobein one-to-one correspondence with the partial actions of G, bothinthecaseof actions on a set, and that of actions as operators on a Hilbert space. In other words, G and S(G) have the same representation theory. We show that S (G) governs the subsemigroup of all closed linear subspaces of a G-graded C ∗-algebra, generated by the grading subspaces. In the special case of finite groups, the maximum number of such subspaces is computed. A “partial ” version of the group C ∗-algebra of a discrete group is introduced. While the usual group C ∗-algebra of finite commutative groups forgets everything but the order of the group, we show that the partial group C ∗-algebra of the two commutative groups of order four, namely Z/4Z and Z/2Z ⊕ Z/2Z, are not isomorphic. 1.

The local structure of a tiling is described in terms of a multiplicative structure on its pattern classes. The groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined. This group furnishes part of the K0-group of the groupoid C ∗-algebra for tilings which reduce to decorations of Z d. The group itself as well as the image of its state is computed for substitution tilings in case the substitution is locally invertible and primitive. This yields in particular the set of possible gap labels predicted by K-theory for Schrödinger operators describing the particle motion in such a tiling. KCL-TH-95-6 1

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

Abstract. There is now a substantial literature on graph C ∗-algebras. Under a locally finite condition on a countable, directed graph, Kumjian, Pask, Raeburn, Renault showed that the C ∗-algebra of the graph can be realized as the C ∗-algebra of the path groupoid, i.e. the groupoid determined by the infinite paths in the graph. In the present paper, we remove the local finiteness requirement. The path groupoid in the general context is obtained through the universal groupoid of a certain inverse semigroup associated with the graph. This inverse semigroup is called the graph inverse semigroup, and graph representations turn out to be just representations of this inverse semigroup. A certain reduction of the universal groupoid gives the path groupoid of the graph, and its C ∗-algebra is isomorphic to the C ∗-algebra of the graph. The unit space of the path groupoid contains the infinite paths of the graph, but also contains some finite paths. We show that, as in the locally finite case, the path groupoid is always amenable, and we give a groupoid proof of a recent theorem of W. Szymanski, characterizing when a graph C ∗-algebra is simple. 1.

Abstract. We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper [CHMM]. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in [CHMM] to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian.