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.