1996/12/23 by D. Guido, Daniele Guido, Guido, D. +3
Mathematics · #58-XX (Primary) 46Lxx (Secondary) #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics #dg-ga #funct-an #math.DG #math.FA #msc:46Lxx #msc:58-XX
paper · pdf · doi:10.48550/arxiv.dg-ga/9612015
LaTeX2e, 39 pages
arxiv created 1996/12/23 · openalex publication_date 1996/12/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A trace on the C^*-algebra A of quasi-local operators on an open manifold is described, based on the results in \citeRoeOpen. It allows a description `a la Novikov-Shubin \citeNS2 of the low frequency behavior of the Laplace-Beltrami operator. The 0-th Novikov-Shubin invariant defined in terms of such a trace is proved to coincide with a metric invariant, which we call asymptotic dimension, thus giving a large scale ``Weyl asymptotics'' relation. Moreover, in analogy with the Connes-Wodzicki result \citeCoCMP,Co,Wo, the asymptotic dimension d measures the singular traceability (at 0) of the Laplace-Beltrami operator, namely we may construct a (type II1) singular trace which is finite on the ^*-bimodule over A generated by Δ-d/2.