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 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)(.) = Tr-omega(.rho(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.