Measurability, spectral densities, and hypertraces in noncommutative geometry

Abstract

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 ${\Omega }_L(\cdot )={\mathrm{Tr}}_{\omega }(\cdot \rho (L))$ 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.