Measurability, Spectral Densities and Hypertracesin Noncommutative Geometry

TL;DR

This paper explores spectral densities and hypertraces in noncommutative geometry, establishing criteria for measurability, unitarity of Dixmier traces, and properties of spectral zeta functions based on spectral multiplicities and eigenvalue growth.

Contribution

It introduces criteria for spectral density measurability, unitarity of Dixmier traces, and analyzes the meromorphic extension of spectral zeta functions in the context of noncommutative geometry.

Findings

Criteria for spectral density measurability and unitarity of Dixmier traces.

Conditions for meromorphic extension and residues of spectral zeta functions.

Hypertrace property of states related to spectral multiplicities and eigenvalue growth.

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 $\Omega_L(\cdot)={\rm Tr\,}_\omega (\cdot\rho(L))$ 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.