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29
Spectral Theory Of Elliptic Operators On NonCompact Manifolds
, 1992
"... 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 ..."
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Cited by 61 (9 self)
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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...
Discreteness of spectrum for the Schrödinger operators on manifolds of bounded geometry
"... 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 ..."
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Cited by 10 (4 self)
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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
A trace on fractal graphs AND THE IHARA ZETA FUNCTION
, 2008
"... 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 MokhtariSharghi have studied zeta functions for infinite graphs acted u ..."
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Cited by 9 (4 self)
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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 MokhtariSharghi 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 selfsimilarity 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
On the geometry of Riemannian manifolds with a Lie structure at infinity
, 2003
"... A manifold with a “Lie structure at infinity” is a noncompact 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 noncompact manifolds ..."
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Cited by 8 (7 self)
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A manifold with a “Lie structure at infinity” is a noncompact 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 noncompact 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 biLipschitz 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 LeviCivita 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
An index for gaugeinvariant operators and the DixmierDouady invariant
 Trans. Amer. Math. Soc
"... Let G → B be a bundle of compact Lie groups acting on a fiber bundle Y → B. In this paper we introduce and study gaugeequivariant Ktheory groups K i G(Y). These groups satisfy the usual properties of the equivariant Ktheory groups, but also some new phenomena arise due to the topological nontriv ..."
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Cited by 8 (1 self)
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Let G → B be a bundle of compact Lie groups acting on a fiber bundle Y → B. In this paper we introduce and study gaugeequivariant Ktheory groups K i G(Y). These groups satisfy the usual properties of the equivariant Ktheory groups, but also some new phenomena arise due to the topological nontriviality of the bundle G → B. As an application, we define a gaugeequivariant 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 gaugeequivariant index as an element of K0(C ∗ (G)), the Ktheory group of the Banach algebra C ∗ (G). We prove that K0(C ∗ (G)) ≃ K 0 G(G) and that the two definitions of the gaugeequivariant index are equivalent. The algebra C ∗ (G) is the algebra of continuous sections of a certain field of C ∗algebras with nontrivial DixmierDouady invariant. The gaugeequivariant Ktheory groups are thus examples of twisted Ktheory groups, which have recently proved themselves useful in the study of RamondRamond fields.
A renormalized index theorem for some complete asymptotically regular metrics: the GaussBonnet theorem
, 2005
"... The GaussBonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the PoincaréEinstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the L²cohomology spaces as well as to carry ..."
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Cited by 8 (1 self)
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The GaussBonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the PoincaréEinstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the L²cohomology 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.
Laplace operators on differential forms over configuration spaces
 J. Geom. Phys
"... 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, ..."
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Cited by 7 (3 self)
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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
Singular traces, dimensions, and NovikovShubin invariants
 Proceedings of the 17th OT Conference, Theta
, 2000
"... ..."
Noncommutative Riemann integration and NovikovShubin invariants for Open Manifolds
, 2001
"... 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 [2 ..."
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Cited by 6 (3 self)
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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 τa.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.
An asymptotic dimension for metric spaces, and the 0th NovikovShubin invariant
"... 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 cl ..."
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Cited by 6 (5 self)
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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 0th NovikovShubin 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.