Results 1 
6 of
6
Dimensions and spectral triples for fractals in R n
 in “Advances in Operator Algebras and Mathematical Physics”, 89–108, Theta Ser. Adv. Math
, 2005
"... Two spectral triples are introduced for a class of fractals in R N. The definitions of noncommutative Hausdorff dimension and noncommutative tangential dimensions, as well as the corresponding Hausdorff and HausdorffBesicovitch functionals considered in [6], are studied for the mentioned fractals e ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
Two spectral triples are introduced for a class of fractals in R N. The definitions of noncommutative Hausdorff dimension and noncommutative tangential dimensions, as well as the corresponding Hausdorff and HausdorffBesicovitch functionals considered in [6], are studied for the mentioned fractals endowed with these spectral triples, showing in many cases their correspondence with classical objects. In particular, for any limit fractal, the HausdorffBesicovitch functionals do not depend on the generalized limit ω. 0 Introduction. In this paper we extend the analysis we made in [6] to fractals in R N, more precisely we define spectral triples for a class of fractals and compare the classical measures, dimensions and metrics with the measures, dimensions and metrics obtained from the spectral triple, in the framework of A. Connes ’ noncommutative
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 ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
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].
unknown title
, 2002
"... Dimensions and singular traces for spectral triples, with applications to fractals ..."
Abstract
 Add to MetaCart
Dimensions and singular traces for spectral triples, with applications to fractals
unknown title
, 2002
"... Dimensions and singular traces for spectral triples, with applications to fractals ..."
Abstract
 Add to MetaCart
Dimensions and singular traces for spectral triples, with applications to fractals
Asymptotic dimension and NovikovShubin invariants for Open Manifolds
, 1996
"... A trace on the C ∗algebra A of quasilocal operators on an open manifold is described, based on the results in [36]. It allows a description à la NovikovShubin [31] of the low frequency behavior of the LaplaceBeltrami operator. The 0th NovikovShubin invariant defined in terms of such a trace is ..."
Abstract
 Add to MetaCart
A trace on the C ∗algebra A of quasilocal operators on an open manifold is described, based on the results in [36]. It allows a description à la NovikovShubin [31] of the low frequency behavior of the LaplaceBeltrami operator. The 0th NovikovShubin invariant defined in terms of such a trace is proved to coincide with a metric invariant, which we call asymptotic dimension, thus giving a large scale “Weyl asymptotics ” relation. Moreover, in analogy with the ConnesWodzicki result [7, 8, 45], the asymptotic dimension d measures the singular traceability (at 0) of the LaplaceBeltrami operator, namely we may construct a (type II1) singular trace which is finite on the ∗bimodule over A generated by ∆ −d/2. 1 Asymptotic dimension and NovikovShubin invariants 2 0 Introduction. The inspiration of this paper came from the idea of Connes ’ [8] of defining the dimension of a noncommutative compact manifold in terms of the Weyl asymptotics,