Diagonality modulo hybrid Lorentz-(p,1) ideals in semifinite factors

Abstract

Let M be a properly infinite semifinite factor and n≥1. For each 1≤i≤n, let Lpi,1(M) be a Lorentz (pi,1)-ideal of M, where p1,…,pn are real numbers satisfying 1≤p1,…,pn<∞ and ∑i=1npi−1≤1. Assuming that the spectral measure of a commuting self-adjoint n-tuple (α(i))i=1n∈Mn is singular, we prove that there exists a commuting self-adjoint diagonal n-tuple (δ(i))i=1n∈Mn such that α(i)−δ(i)∈Lpi,1(M), 1≤i≤n. Moreover, max1≤i≤n⁡max⁡{‖α(i)−δ(i)‖Lpi,1(M),‖α(i)−δ(i)‖M} can be arbitrarily small. This extends an earlier result due to Voiculescu.