The Quasicentral Modulus Associated with a Class of Nonself-similar Fractals

TL;DR

This work extends Voiculescu's framework for the quasicentral modulus to tuples of commuting self-adjoint operators whose spectral measures are absolutely continuous with respect to generalized Hausdorff measures generated by gauge functions $f$. It identifies maximal obstruction ideals in this fractal-measure setting, proves an ampliation homogeneity for the associated moduli, and derives an exact formula linking $(k_ ho( au))^s$ to the $H_f$-weighted multiplicity on the spectrum. The results rely on constructing symmetric Cantor-type fractals $C_f$, establishing spectral decompositions relative to $H_f$, and reducing to multiplication by coordinates on $L^2$ spaces. The findings provide a precise criterion for diagonalization modulo obstruction ideals in nonself-similar fractal contexts and generalize Voiculescu's integral formula to a broader class of measures with practical implications for operator-theoretic diagonalization problems on fractal spectra.

Abstract

We discuss an extension to Voiculescu's formula for the quasicentral modulus of a tuple of commuting, self-adjoint operators with spectral measure absolutely continuous with respect to a generalized Hausdorff measure. These Hausdorff measures are defined by gauge functions which are not power functions and are supported on nonself-similar fractals.

Theorems & Definitions (35)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 3.1
• proof
• Lemma 3.2