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Pré-Publication, Document De Travail Année : 2021

Measurability, Spectral Densities and Hypertracesin Noncommutative Geometry

Résumé

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.

Dates et versions

hal-04012371 , version 1 (02-03-2023)

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Fabio E. G. Cipriani, Jean-Luc Sauvageot. Measurability, Spectral Densities and Hypertracesin Noncommutative Geometry. 2021. ⟨hal-04012371⟩
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