Anti-norms on Finite von Neumann Algebras

Abstract

As the reversed version of usual symmetric norms, we introduce the notion of symmetric anti-norms $\Vert \cdot {\Vert }_{!}$ 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 $\Vert \psi (A+B){\Vert }_{!}\ge \Vert \psi (A){\Vert }_{!}+\Vert \psi (B){\Vert }_{!}$ for a wide class of functions $\psi$.