Measurability, spectral densities, and hypertraces in noncommutative geometry.

We introduce, in the dual Macaev ideal of compact operators of a Hilbert space, the spectral weight ρ(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 (ρ(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 ζ-function ζ_L of a spectral density are provided under summability conditions on spectral multiplicities. The hypertrace property of the states Ω_L(·) = Tr_ω(·ρ(L)) on the norm closure of the Lipschitz algebra A_L follows if the relative multiplicities of L vanish faster than its spectral gaps or if N_L is asymptotically regular. [ABSTRACT FROM AUTHOR]