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 ...

We prove a version of the Atiyah-Segal completion theorem for proper actions of an infinite discrete group G. More precisely, for any finite proper G-CW-complex X, K (EG\Theta GX) is the completion of K G (X) with respect to a certain ideal. We also show, for such G and X, that KG (X) can be defined as the Grothendieck group of the monoid of G-vector bundles over X. Key words: K-theory, proper actions, vector bundles 1991 mathematics subject classification: primary 55N91, secondary 19L47 Let G be any discrete group. For such G, a G-CW-complex is a CW-complex with G- action which permutes the cells, such that an element g 2 G sends a cell to itself only by the identity map. A G-CW-complex X is proper if all of its isotropy subgroups have finite order, and is finite if it is made up of finitely many orbits of cells. A G-CW-pair is a pair of G-spaces (X; A), where X is a G-CW-complex and A is a G-invariant subcomplex. The main results of this paper are Theorems 3.2 and 4.3 below...

Abstract. We define the Brauer group Br(G) of a locally compact groupoid G to be the set of Morita equivalence classes of pairs (A, α) consisting of an elementary C ∗-bundle A over G (0) satisfying Fell’s condition and an action α of G on A by ∗-isomorphisms. When G is the transformation groupoid X × H, then Br(G) is the equivariant Brauer group BrH(X). In addition to proving that Br(G) is a group, we prove three isomorphism results. First we show that if G and H are equivalent groupoids, then Br(G) and Br(H) are isomorphic. This generalizes the result that if G and H are groups acting freely and properly on a space X, say G on the left and H on the right then BrG(X/H) and BrH(G\X) are isomorphic. Secondly we show that the subgroup Br0(G) of Br(G) consisting of classes [A, α] with A having trivial Dixmier-Douady invariant is isomorphic to a quotient E(G) of the collection Tw(G) of twists over G. Finally we prove that Br(G) is isomorphic to the inductive limit Ext(G, T) of the groups E(G X) where X varies over all principal G spaces X and G X is the imprimitivity groupoid associated to X. 1.

An orbifold is a singular space which is locally modeled on the quotient of a smooth manifold by a smooth action of a finite group. It appears naturally in geometry and topology when group actions on manifolds are involved and the stabilizer of each fixed point is finite. The concept of an orbifold was first introduced by Satake under the name “V-manifold ” in a paper where he also extended the basic differential geometry to his newly defined singular spaces (cf. [Sa]). The local structure of an orbifold – being the quotient of a smooth manifold by a finite group action – was merely used as some “generalized smooth structure”. A different aspect of the local structure was later recognized by Thurston, who gave the name “orbifold ” and introduced an important concept – the fundamental group of an orbifold (cf. [Th]). In 1985, physicists Dixon, Harvey, Vafa and Witten studied string theories on Calabi-Yau orbifolds (cf. [DHVW]). An interesting discovery in their paper was the prediction that a certain physicist’s Euler number of the orbifold must be equal to the Euler number of any of its crepant resolutions. This was soon related to the so called McKay correspondence in mathematics (cf. [McK]). Later developments include stringy Hodge numbers (cf. [Z], [BD]), mirror symmetry of

Let X be a simply connected space and K be any field. The normalized singular cochains N (X ; K) admit a natural strongly homotopy commutative algebra structure, which induces a natural product on the Hochschild homology HH N X of the space X . We prove that, endowed with this product, HH N X is isomorphic to the cohomology algebra of the free loop space of X with coefficients in K. We also show how to construct a simpler Hochschild complex which allows direct computation.

When a Hamiltonian system has a "Kinetic + Potential" structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure, so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure.

JOP- Vol. 6, Do 2. 1989

Let A be a fixed C*-algebra. In an arbitrary finitely generated projective A-module V ⊆ An, a spherical tight A-frame is a set of of k, k> n, elements f1,...,fk such that the associated matrix F = [f1,...,fk] up-to a constant multiple is a partial isometry of the Hilbert structure on the projective finitely generated A-module V. The space FA k,n of all such A-frames form a C*-algebra, generated by a system of partial isometries and the structure of such C*-algebras are well described, especially in the case A = R or C: The main result of K. Dykema and N. Strawn for these cases are generalized to our general projective finitely generated Hilbert A-module case. This generalization gives the possibility to study the universal classifying space. Keywords: spherical tight frame, K-theory

On the cohomology algebra of free loop spaces. by