A logarithmically monotone symmetric norm which is not fully symmetric

Abstract

We construct an explicit symmetric Banach norm on a function space over $(0,\infty )$ which is monotone with respect to logarithmic submajorization but is not strongly symmetric, and hence is not fully symmetric. This answers a question raised by Dodds, Dodds, Sukochev, and Zanin. We also construct a strongly symmetric Banach function space over $(0,\infty )$ which is monotone with respect to logarithmic submajorization and admits no equivalent fully symmetric norm.