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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].
HOMOGENIZATION ON LATTICES: SMALL PARAMETER LIMITS, HMEASURES, AND DISCRETE WIGNER MEASURES
"... Abstract. We fully characterize the smallparameter limit for a class of lattice models with twoparticle long or short range interactions with no \exchange energy. " One of the problems we consider is that of characterizing the continuum limit of the classical magnetostatic energy of a sequence of m ..."
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Cited by 1 (0 self)
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Abstract. We fully characterize the smallparameter limit for a class of lattice models with twoparticle long or short range interactions with no \exchange energy. " One of the problems we consider is that of characterizing the continuum limit of the classical magnetostatic energy of a sequence of magnetic dipoles on a Bravais lattice, (letting the lattice parameter tend to zero). In order to describe the smallparameter limit, we use discrete Wigner transforms to transform the storedenergy which is given by the double convolution of a sequence of (dipole) functions on a Bravais lattice with a kernel, homogeneous of degree with N with the cancellation property, as the lattice parameter tends to zero. By rescaling and using Fourier methods, discrete Wigner transforms in particular, to transform the problem to one on the torus, we are able to characterize the smallparameter limit of the energy depending on whether the dipoles oscillate on the scale of the lattice, oscillate on a much longer lengthscale, or converge strongly. In the case where> N, the result is simple and can be characterized by anintegral with respect to the Wigner measure limit on the torus. In the case where = N, oscillations essentially on the scale of the lattice must be separated from oscillations essentially onamuch longer lengthscale in order to characterize the energy in terms of the Wigner measure limit on the torus, an Hmeasure limit, and the limiting magnetization. We show that the classical
DOUBLEDUAL TYPES OVER THE BANACH SPACE C(K)
, 2005
"... Let K be a compact Hausdorff space and C(K) the Banach space of all realvalued continuous functions on K, with the supnorm. Types over C(K) (in the sense of Krivine and Maurey) can be uniquely represented by pairs (ℓ,u) of bounded realvalued functions on K, whereℓ is lower semicontinuous, u is up ..."
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Let K be a compact Hausdorff space and C(K) the Banach space of all realvalued continuous functions on K, with the supnorm. Types over C(K) (in the sense of Krivine and Maurey) can be uniquely represented by pairs (ℓ,u) of bounded realvalued functions on K, whereℓ is lower semicontinuous, u is upper semicontinuous, ℓ ≤ u, andℓ(x) = u(x) for all isolated points x of K. A condition that characterizes the pairs (ℓ,u) that represent doubledual types over C(K)isgiven. 1. Statement of the main theorem The concept of type over a Banach space E was first introduced by Krivine and Maurey [7] in the context of separable Banach spaces. The reader is referred to Garling’s monograph [4] for more details. We consider general, not necessarily separable Banach spaces. Let E be a Banach space. For every x ∈ E, we define a function τx: E → R by letting τx(y) = ‖x + y ‖ for all y ∈ E. Definition 1.1. A function τ: E → R is a type over E if τ is in the closure (with respect to the topology of pointwise convergence) of the set {τx: x ∈ E}. The definition given here is equivalent to the definition given in [1]. That is, τ is a type over E if and only if there exists an ultrafilter � over an infinite index set λ and a bounded family of elements (xα)α∈λ in E such that τ(y) = limα∈ � ‖xα + y ‖ for all y ∈ E.Thereader is referred to [5] for more details regarding the choice of the ultrafilter. Throughout, we let K be a compact Hausdorff topological space. The topology on K is denoted by Ω. Weletℓ∞(K) denote the Banach lattice of bounded realvalued functions on K equipped with the supnorm. For f,g ∈ ℓ∞(K), the lattice ordering is defined pointwise. An sc pair (semicontinuous pair) is a pair of functions (ℓ,u)fromℓ∞(K)suchthatℓ is lower semicontinuous (lsc), u is upper semicontinuous (usc), ℓ ≤ u, andℓ(x) = u(x) for all isolated points x ∈ K. The Banach space of continuous realvalued functions on K with supnorm is denoted by C(K). The constant function with value 1 is denoted by 1.