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An asymptotic dimension for metric spaces, and the 0th NovikovShubin invariant
"... A nonnegative number d∞, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a cl ..."
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Cited by 6 (5 self)
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A nonnegative number d∞, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a class of open manifolds with bounded geometry the asymptotic dimension coincides with the 0th NovikovShubin number α0 defined in a previous paper [D. Guido, T. Isola, J. Funct. Analysis, 176 (2000)]. Thus the dimensional interpretation of α0 given in the mentioned paper in the framework of noncommutative geometry is established on metrics grounds. Since the asymptotic dimension of a covering manifold coincides with the polynomial growth of its covering group, the stated equality generalises to open manifolds a result by Varopoulos. 0. Introduction.
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
unknown title
, 2000
"... An asymptotic dimension for metric spaces, and the 0th NovikovShubin invariant ..."
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An asymptotic dimension for metric spaces, and the 0th NovikovShubin invariant