Quasicentral modulus and self-similar sets: a supplementary result to Voiculescu's work

TL;DR

This paper extends Voiculescu's formula for the quasicentral modulus to a broader class of self-similar sets satisfying the open set condition, including fractals like the Sierpinski gasket and carpet.

Contribution

It provides bounds for the quasicentral modulus for self-similar sets and confirms Voiculescu's formula applies to these fractals.

Findings

Voiculescu's formula holds for certain self-similar sets.

Established bounds for the quasicentral modulus on these sets.

Includes classical fractals like Sierpinski gasket and carpet.

Abstract

In his recent work, Voiculescu generalized his remarkable formula for the quasicentral modulus of a commuting $n$-tuple of hermitian operators with respect to the $(n,1)$-Lorentz ideal to the case where its spectrum is contained in a Cantor-like self-similar set in a certain class. In this note, we treat general self-similar sets satisfying the open set condition, and obtain lower and upper bounds of the quasicentral modulus. Our proof shows that Voiculescu's formula holds for a class of self-similar sets including the Sierpinski gasket and the Sierpinski carpet.