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

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.