We introduce spectral weights ρ(L) of positive, self-adjoint operators L having dis-
crete spectrum away from zero. We provide criteria for measurability and unitarity of its Dixmier traces (ρ(L) are then called spectral densities) in terms of the growth of spectral multiplicities of L or in terms of the asymptotic continuity of the eigenvalue counting function N_L of L. Existence of meromorphic extensions and residues of the ζ-function of spectral densities are provided under summability conditions on spectral multiplicities. The hypertrace property of the states ΩL(·) = Tr ω(·ρ(L)) on the norm closure of the Lipschitz algebra of L follows if the relative multiplicities of L vanish faster then its spectral gaps or if N_L is asymptotically regular. Examples include countable discrete groups, Toeplitz C*-algebra, algebra of 0-order ΨDO, Euclidean domains of infinite volume, Voiculescu spectral triple associated to Hilbert space filtrations.
This is joint work with J.-L. Sauvageot.