1998/09/08 by Daniele Guido, Guido, Daniele, Tommaso Isola +1
Mathematics · #58-XX (Primary) 46Lxx (Secondary) #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Operator Algebras (math.OA) #math.DG #math.OA #msc:46Lxx #msc:58-XX
paper · pdf · doi:10.48550/arxiv.math/9809040
43 pages, LaTex2e
arxiv created 1998/09/08 · openalex publication_date 1998/09/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.