Anti-norms on Finite von Neumann Algebras

Abstract

As the reversed version of usual symmetric norms, we introduce the notion of symmetric anti-norms $|\cdot|!$ defined on the positive operators affiliated with a finite von Neumann algebra with a finite normal trace. Related to symmetric anti-norms, we develop majorization theory and superadditivity inequalities of the form $|\psi(A+B)|!\ge|\psi(A)|!+|\psi(B)|!$ for a wide class of functions $\psi$.