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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.
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 4 (4 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 L²Betti numbers
, 2006
"... 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, L²Betti numbers and NovikovShubin numbers are defined for such complexes ..."
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Cited by 2 (2 self)
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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, L²Betti numbers and NovikovShubin numbers are defined for such complexes in terms of the trace. In particular a relation involving the EulerPoincaré characteristic is proved. L²Betti and NovikovShubin numbers are computed for some selfsimilar complexes arising from selfsimilar fractals.