preliminaries Let H be a complex Hilbert space, A a densely dened linear operator in H (the domain of A will be denoted D(A)). Suppose that A has a closure A or, equivalently, that the adjoint operator A is densely dened (see e.g. [32]). We shall denote by GA the graph of A i.e. the set of pairs fu; Aug; u 2 D(A). Then G A = GA , i.e. the graph of A is the closure of the graph of A. Moreover A = A = (A ) . Now let A + be another densely dened linear operator in H. DEFINITION 1.1. A + is called formally adjoint to A if (1:1) (Au; v) = (u; A + v); u 2 D(A); v 2 D(A + ); where (; ) is the scalar product in H. If A = A + then A is called symmetric or formally self{adjoint. Note that since A; A + are densely dened, both A and A + have closures. DEFINITION 1.2. Let A; A + be as in Denition 1.1. Then the minimal and the maximal operator for A are dened as follows: A min = A = A ; A max = (A + ) : Note that both A min and A max are...

this paper we will assume that (M; g) is a manifold of bounded geometry, i.e. the following two conditions are satised: (a) r inj > 0 where r inj is the radius of injectivity of M ; (b) jr

Starting with Ihara’s work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and Mokhtari-Sharghi have studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, i.e. they enjoy a self-similarity property. We define a zeta function for these graphs and, using the machinery of operator algebras, we prove a determinant formula, which relates the zeta function with the Laplacian of the graph. We also prove functional equations, and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs. The Ihara zeta function, originally associated to certain groups and then combinatorially

Abstract. A manifold with a “Lie structure at infinity ” is a non-compact manifold M0 whose geometry is described by a compactification to a manifold with corners M and a Lie algebraV of vector fields on M subject to constraints only on M �M0. This definition recovers several classes of non-compact manifolds that were studied before: manifolds with cylindrical ends, manifolds that are Euclidean at infinity, conformally compact manifolds, and others. It hence provides a unified setting for the study of these classes of manifolds and of their geometric differential operators. The Lie structure at infinity on M0 determines a complete metric on M0 up to bi-Lipschitz equivalence. This leads to the natural problem of understanding the Riemannian geometry of these manifolds, which is the main question addressed in this paper. We prove, for example, that on a manifold with a Lie structure at infinity the curvature tensor and its covariant derivatives are bounded, by extending the Levi-Civita connection to an A ∗-valued connection where the bundle A is uniquely determined by the Lie algebra V. We study a generalization of the geodesic spray

Let G → B be a bundle of compact Lie groups acting on a fiber bundle Y → B. In this paper we introduce and study gauge-equivariant K-theory groups K i G(Y). These groups satisfy the usual properties of the equivariant K-theory groups, but also some new phenomena arise due to the topological non-triviality of the bundle G → B. As an application, we define a gauge-equivariant index for a family of elliptic operators (Pb)b∈B invariant with respect to the action of G → B, which, in this approach, is an element of K 0 G(B). We then give another definition of the gauge-equivariant index as an element of K0(C ∗ (G)), the K-theory group of the Banach algebra C ∗ (G). We prove that K0(C ∗ (G)) ≃ K 0 G(G) and that the two definitions of the gauge-equivariant index are equivalent. The algebra C ∗ (G) is the algebra of continuous sections of a certain field of C ∗-algebras with non-trivial Dixmier-Douady invariant. The gauge-equivariant K-theory groups are thus examples of twisted K-theory groups, which have recently proved themselves useful in the study of Ramond-Ramond fields.

Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation is given. 2000 AMS Mathematics Subject Classification. 60G57, 58A10Contents

Abstract. The Gauss-Bonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the Poincaré-Einstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the L 2-cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod x m, the finite time supertrace of the heat kernel on conformally compact manifolds is shown to renormalize independently of the choice of special boundary defining function.

Abstract not found

Given a C ∗-algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [26], and show that A R is a C ∗-algebra, and τ extends to a semicontinuous semifinite trace on A R. Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A ′′ and can be approximated in measure by operators in A R, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a τ-a.e. bimodule on A R, denoted by A R, and such bimodule contains the functional calculi of selfadjoint elements of A R under unbounded Riemann measurable functions. Besides, τ extends to a bimodule trace on A R.

A nonnegative number d∞, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a class of open manifolds with bounded geometry the asymptotic dimension coincides with the 0-th Novikov-Shubin number α0 defined in a previous paper [D. Guido, T. Isola, J. Funct. Analysis, 176 (2000)]. Thus the dimensional interpretation of α0 given in the mentioned paper in the framework of noncommutative geometry is established on metrics grounds. Since the asymptotic dimension of a covering manifold coincides with the polynomial growth of its covering group, the stated equality generalises to open manifolds a result by Varopoulos. 0. Introduction.