2,467 research outputs found

### An asymptotic dimension for metric spaces, and the 0-th Novikov-Shubin invariant

A nonnegative number d_infinity, 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 0-th Novikov-Shubin number alpha_0 defined previously
(math.OA/9802015, cf. also math.DG/0110294). Thus the dimensional
interpretation of alpha_0 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.Comment: 17 pages, to appear on the Pacific Journal of Mathematics. This paper
roughly corresponds to the third section of the unpublished math.DG/980904

### Novikov-Shubin invariants and asymptotic dimensions for open manifolds

The Novikov-Shubin numbers are defined for open manifolds with bounded
geometry, the Gamma-trace of Atiyah being replaced by a semicontinuous
semifinite trace on the C*-algebra of almost local operators. It is proved that
they are invariant under quasi-isometries and, making use of the theory of
singular traces for C*-algebras developed in math/9802015, they are interpreted
as asymptotic dimensions since, in analogy with what happens in Connes'
noncommutative geometry, they indicate which power of the Laplacian gives rise
to a singular trace. Therefore, as in geometric measure theory, these numbers
furnish the order of infinitesimal giving rise to a non trivial measure. The
dimensional interpretation is strenghtened in the case of the 0-th
Novikov-Shubin invariant, which is shown to coincide, under suitable geometric
conditions, with the asymptotic counterpart of the box dimension of a metric
space. Since this asymptotic dimension coincides with the polynomial growth of
a discrete group, the previous equality generalises a result by Varopoulos for
covering manifolds. This paper subsumes dg-ga/9612015. In particular, in the
previous version only the 0th Novikov-Shubin number was considered, while here
Novikov-Shubin numbers for all p are defined and studied.Comment: 43 pages, LaTex2

### A Remark on Gelfand Duality for Spectral Triples

We present a duality between the category of compact Riemannian spin
manifolds (equipped with a given spin bundle and charge conjugation) with
isometries as morphisms and a suitable "metric" category of spectral triples
over commutative pre-C*-algebras. We also construct an embedding of a
"quotient" of the category of spectral triples introduced in
arXiv:math/0502583v1 into the latter metric category. Finally we discuss a
further related duality in the case of orientation and spin-preserving maps
between manifolds of fixed dimension.Comment: 15 pages, AMS-LaTeX2e, results unchanged, several improvements in the
exposition, appendix adde

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