We introduce, in the dual Macaev ideal of compact operators of a Hilbert space, the spectral weight of a positive, self-adjoint operator having discrete spectrum away from zero. We provide criteria for its measurability and unitarity of its Dixmier traces ( is then called spectral density) in terms of the growth of the spectral multiplicities of or in terms of the asymptotic continuity of the eigenvalue counting function . Existence of meromorphic extensions and residues of the -function of a spectral density are provided under summability conditions on spectral multiplicities. The hypertrace property of the states on the norm closure of the Lipschitz algebra follows if the relative multiplicities of vanish faster than its spectral gaps or if is asymptotically regular.
Cite this article
Fabio E.G. Cipriani, Jean-Luc Sauvageot, Measurability, spectral densities, and hypertraces in noncommutative geometry. J. Noncommut. Geom. 17 (2023), no. 4, pp. 1437–1468DOI 10.4171/JNCG/511