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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 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.
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].
A Semicontinuous Trace for Almost Local Operators on an Open Manifold
, 2001
"... A semicontinuous semifinite trace is constructed on the C*algebra ..."
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Cited by 3 (3 self)
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A semicontinuous semifinite trace is constructed on the C*algebra
A C ∗ algebra of geometric operators on selfsimilar CWcomplexes. Novikov–Shubin and L 2 Betti numbers, preprint
, 2006
"... Abstract. A class of CWcomplexes, called selfsimilar complexes, is introduced, together with C ∗algebras Aj of operators, endowed with a finite trace, acting on squaresummable cellular jchains. Since the Laplacian ∆j belongs to Aj, L 2Betti numbers and NovikovShubin numbers are defined for su ..."
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Cited by 2 (2 self)
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Abstract. A class of CWcomplexes, called selfsimilar complexes, is introduced, together with C ∗algebras Aj of operators, endowed with a finite trace, acting on squaresummable cellular jchains. Since the Laplacian ∆j belongs to Aj, L 2Betti numbers and NovikovShubin numbers are defined for such complexes in terms of the trace. In particular a relation involving the EulerPoincaré characteristic is proved. L 2Betti and NovikovShubin numbers are computed for some selfsimilar complexes arising from selfsimilar fractals. 1. Introduction. In this paper we address the question of the possibility of extending the definition of some L 2invariants, like the L 2Betti numbers and NovikovShubin numbers, to geometric structures which are not coverings of compact spaces. The first attempt in this sense is due to John Roe [29], who defined a trace
Topology of covers and spectral theory of geometric operators
 Proceedings Conf. on Khomology
, 1993
"... operators ..."
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 ..."
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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,