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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.
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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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