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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 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.
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