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Tangential dimensions I. Metric spaces
"... Abstract. Pointwise tangential dimensions are introduced for metric spaces. Under regularity conditions, the upper, resp. lower, tangential dimensions of X at x can be defined as the supremum, resp. infimum, of box dimensions of the tangent sets, a la Gromov, of X at x. Our main purpose is that of i ..."
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Abstract. Pointwise tangential dimensions are introduced for metric spaces. Under regularity conditions, the upper, resp. lower, tangential dimensions of X at x can be defined as the supremum, resp. infimum, of box dimensions of the tangent sets, a la Gromov, of X at x. Our main purpose is that of introducing a tool which is very sensitive to the “multifractal behaviour at a point ” of a set, namely which is able to detect the “oscillations ” of the dimension at a given point. In particular we exhibit examples where upper and lower tangential dimensions differ, even when the local upper and lower box dimensions coincide. Tangential dimensions can be considered as the classical analogue of the tangential dimensions for spectral triples introduced in [7], in the framework of Alain Connes ’ noncommutative geometry [4]. 1. Introduction. Dimensions can be seen as a tool for measuring the nonregularity, or fractality, of a given object. Nonintegrality of the dimension is a first sign of nonregularity. A second kind of nonregularity is related to the fact that the dimension is not a
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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A semicontinuous semifinite trace is constructed on the C*algebra
Tangential dimensions for metric spaces and measures
"... Notions of (pointwise) tangential dimension are considered, both for subsets and measures of R N. Under regularity conditions (volume doubling), the upper resp. lower dimension at a point x of a measure µ can be defined as the supremum, resp. infimum, of local dimensions of the measures tangent to µ ..."
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Cited by 3 (3 self)
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Notions of (pointwise) tangential dimension are considered, both for subsets and measures of R N. Under regularity conditions (volume doubling), the upper resp. lower dimension at a point x of a measure µ can be defined as the supremum, resp. infimum, of local dimensions of the measures tangent to µ at x. Moreover, we introduce and study the notion of tangent space of a closed subset X of R N at x as the collection of its tangent sets at x, defined as suitable (Attouch Wets) limits of dilations of X around the point x. Then, under regularity conditions, the upper, resp. lower, tangential dimensions of X at x can be defined as the supremum, resp. infimum, of box dimensions of the sets tangent to X at x. Our main purpose is that of introducing a tool which is very sensitive to the ”multifractal behaviour at a point ” of a set, resp. measure, namely which is able to detect the ”oscillations ” of the dimension at a given point, even when the local dimension exists, namely local upper and lower dimensions coincide. These definitions are tested on a class of fractals, which we call translation fractals, where they can be explicitly calculated. In these cases the tangential dimensions of the fractal coincide with the tangential dimensions of an associated invariant measure, and they are constant, i.e. do not depend on the point. However, upper and lower dimensions may differ. Moreover, on these fractals, these quantities coincide with their noncommutative analogues, defined in a previous paper [9], in the framework of Alain Connes ’ noncommutative geometry.
The problem of completeness for GromovHausdorff metrics on C ∗algebras ∗
, 2005
"... It is proved that the family of equivalence classes of Lipnormed C ∗algebras introduced by M. Rieffel, up to complete order isomorphisms preserving the Lipseminorm, is not complete w.r.t. the matricial quantum GromovHausdorff distance introduced by D. Kerr. This is shown by exhibiting a Cauchy s ..."
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It is proved that the family of equivalence classes of Lipnormed C ∗algebras introduced by M. Rieffel, up to complete order isomorphisms preserving the Lipseminorm, is not complete w.r.t. the matricial quantum GromovHausdorff distance introduced by D. Kerr. This is shown by exhibiting a Cauchy sequence whose limit, which always exists as an operator system, is not completely order isomorphic to any C ∗algebra. Conditions ensuring the existence of a C ∗structure on the limit are considered, making use of the notion of ultraproduct. More precisely, a necessary and sufficient condition is given for the existence, on the limiting operator system, of a C ∗product structure inherited from the approximating C ∗algebras. Such condition can be considered as a generalisation of the fLeibniz conditions introduced by Kerr and Li. Furthermore, it is shown that our condition is not necessary for the existence of a C ∗structure tout court, namely there are cases in which the limit is a C ∗algebra, but the C ∗structure is not inherited. 1
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"... Tangential dimensions. I. Metric spaces. (English summary) ..."