Anti-norms on finite von Neumann algebras

TL;DR

This paper introduces symmetric anti-norms on positive operators in finite von Neumann algebras, develops related majorization theory, and establishes superadditivity inequalities for a broad class of functions.

Contribution

It defines symmetric anti-norms on finite von Neumann algebras and explores their properties, including majorization and superadditivity inequalities, extending the theory of operator norms.

Findings

Established superadditivity inequalities for symmetric anti-norms.

Developed majorization theory related to anti-norms.

Provided a framework for analyzing 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 $\psi$.