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14
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 9 (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
"... ..."
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.
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].
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
Symmetry, Integrability and Geometry: Methods and Applications Measure Theory in Noncommutative Spaces ⋆
"... doi:10.3842/SIGMA.2010.072 Abstract. The integral in noncommutative geometry (NCG) involves a nonstandard trace called a Dixmier trace. The geometric origins of this integral are well known. From a measuretheoretic view, however, the formulation contains several difficulties. We review results con ..."
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doi:10.3842/SIGMA.2010.072 Abstract. The integral in noncommutative geometry (NCG) involves a nonstandard trace called a Dixmier trace. The geometric origins of this integral are well known. From a measuretheoretic view, however, the formulation contains several difficulties. We review results concerning the technical features of the integral in NCG and some outstanding problems in this area. The review is aimed for the general user of NCG. Key words: Dixmier trace; singular trace; noncommutative integration; noncommutative
TYPE II NON COMMUTATIVE GEOMETRY.
, 2003
"... Abstract. 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 t ..."
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Abstract. 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. Contents
SUMS OF COMMUTATORS IN IDEALS AND MODULES OF TYPE II FACTORS
, 2003
"... Abstract. Let M be a factor of type II ∞ or II1 having separable predual and let M be the algebra of affiliated τ–measureable operators. We characterize the commutator space [I, J] for sub–(M, M)–bimodules I and J of M. ..."
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Abstract. Let M be a factor of type II ∞ or II1 having separable predual and let M be the algebra of affiliated τ–measureable operators. We characterize the commutator space [I, J] for sub–(M, M)–bimodules I and J of M.