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TYPE II NON COMMUTATIVE GEOMETRY. I. Dixmier Trace In Von Neumann Algebras
, 2003
"... We define the notion of Connesvon Neumann spectral triple and consider the associated index problem. We compute the analytic ChernConnes character of such a generalized spectral triple and prove the corresponding local formula for its Hochschild class. This formula involves the Dixmier trace for ..."
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Cited by 35 (3 self)
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We define the notion of Connesvon Neumann spectral triple and consider the associated index problem. We compute the analytic ChernConnes character of such a generalized spectral triple and prove the corresponding local formula for its Hochschild class. This formula involves the Dixmier trace for II∞ von Neumann algebras. In the case of foliations, we identify this Dixmier trace with the corresponding measured Wodzicki residue.
Fractals in Noncommutative Geometry
 in the Proceedings of the Conference ”Mathematical Physics in Mathematics and Physics”, Siena 2000, Edited by R. Longo, Fields Institute Communications
, 2001
"... Abstract. To any spectral triple (A, D, H) a dimension d is associated, in analogy with the Hausdorff dimension for metric spaces. Indeed d is the unique number, if any, such that D  −d has non trivial logarithmic Dixmier trace. Moreover, when d ∈ (0, ∞), there always exists a singular trace which ..."
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Cited by 11 (6 self)
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Abstract. To any spectral triple (A, D, H) a dimension d is associated, in analogy with the Hausdorff dimension for metric spaces. Indeed d is the unique number, if any, such that D  −d has non trivial logarithmic Dixmier trace. Moreover, when d ∈ (0, ∞), there always exists a singular trace which is finite nonzero on D  −d, giving rise to a noncommutative integration on A. Such results are applied to fractals in R, using Connes ’ spectral triple, and to limit fractals in R n, a class which generalises selfsimilar fractals, using a new spectral triple. The noncommutative dimension or measure can be computed in some cases. They are shown to coincide with the (classical) Hausdorff dimension and measure in the case of selfsimilar fractals. 1 Introduction. This paper is both a survey and an announcement of results concerning singular traces on B(H), and their application to the study of fractals in the framework of
Singular traces, dimensions, and NovikovShubin invariants
 Proceedings of the 17th OT Conference, Theta
, 2000
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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 7 (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.
Noncommutative Riemann integration and singular traces for C ∗  algebras
"... 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 [1 ..."
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Cited by 4 (4 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 [16], 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 improper Riemann integration. Unbounded Riemann measurable operators form a τa.e. bimodule on A R, denoted by AR, 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 AR. As type II1 singular traces for a semifinite von Neumann algebra M with a normal semifinite faithful (nonatomic) trace τ have been defined as traces on M − Mbimodules of unbounded τmeasurable operators [5], type II1 singular traces for a C ∗algebra A with a semicontinuous semifinite (nonatomic) trace τ are defined here as traces on A − Abimodules of unbounded Riemann measurable operators (in AR) for any faithful representation of A. An application of singular traces for C ∗algebras is contained in [6].
Spectral triples on the Sierpinski gasket
 AMS Meeting ‘Analysis, Probability and Mathematical Physics on Fractals’, Cornell U
, 2011
"... Abstract. We construct a 2parameter family of spectral triples for the Sierpinski Gasket K. For suitable values of the parameters we determine the dimensional spectrum and recover the Hausdorff measure of K in terms of the residue of the functional a → tr(a D−s) at the abscissa of convergence d, ..."
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Cited by 4 (1 self)
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Abstract. We construct a 2parameter family of spectral triples for the Sierpinski Gasket K. For suitable values of the parameters we determine the dimensional spectrum and recover the Hausdorff measure of K in terms of the residue of the functional a → tr(a D−s) at the abscissa of convergence d, which coincides with the Hausdorff dimension of the fractal. We determine the associated Connes ’ distance showing that it is biLipschitz equivalent to a suitable root of the Euclidean metric of the plane, and show that the pairing of the associated Fredholm module with (odd) Ktheory is nontrivial. We recover also the unique, standard Dirichlet form on K, as the residue of the functional a → tr(D−s/2[D, a]2 D−s/2) at the abscissa of convergence δ, which we call the energy dimension. The fact that the volume dimension differs from the energy dimension, d 6 = δ, reflects the fact that on K energy and volume are distributed singularly. 1.
On the domain of singular traces
 Int. J. Math
"... After the introduction of spectral triples in Alain Connes ’ Noncommutative Geometry, singular traces on B(H) became a quite popular tool in operator algebras. With the aim of classifying them some papers have been written ([3],[8],[1], see also [4] for nonpositive traces), addressing in particular ..."
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Cited by 3 (2 self)
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After the introduction of spectral triples in Alain Connes ’ Noncommutative Geometry, singular traces on B(H) became a quite popular tool in operator algebras. With the aim of classifying them some papers have been written ([3],[8],[1], see also [4] for nonpositive traces), addressing in particular the question