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Measurability, Spectral Densities and Hypertracesin Noncommutative Geometry

We introduce, in the dual Macaev ideal of compact operators of a Hilbert space, the spectral weight \(\rho(L)\) of a positive, self-adjoint operator \(L\) having discrete spectrum away from zero. We provide criteria for its measurability and unitarity of its Dixmier traces (\(\rho(L)\) is then calle...

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Bibliographic Details
Published in:arXiv.org 2021-11
Main Authors: Cipriani, Fabio E G, Sauvageot, Jean-Luc
Format: Article
Language:English
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Summary:We introduce, in the dual Macaev ideal of compact operators of a Hilbert space, the spectral weight \(\rho(L)\) of a positive, self-adjoint operator \(L\) having discrete spectrum away from zero. We provide criteria for its measurability and unitarity of its Dixmier traces (\(\rho(L)\) is then called spectral density) in terms of the growth of the spectral multiplicities of \(L\) or in terms of the asymptotic continuity of the eigenvalue counting function \(N_L\). Existence of meromorphic extensions and residues of the \(\zeta\)-function \(\zeta_L\) of a spectral density are provided under summability conditions on spectral multiplicities. The hypertrace property of the states \(\Omega_L(\cdot)={\rm Tr\,}_\omega (\cdot\rho(L))\) on the norm closure of the Lipschitz algebra \(\mathcal{A}_L\) follows if the relative multiplicities of \(L\) vanish faster than its spectral gaps or if \(N_L\) is asymptotically regular.
ISSN:2331-8422