Inequalities for trace on $\tau$-measurable operators

TL;DR

This paper extends classical trace inequalities to $ au$-measurable operators in semifinite von Neumann algebras, including Clarkson inequalities and a parallelogram law, broadening the scope of operator inequalities.

Contribution

It introduces new inequalities for $ au$-measurable operators, generalizing known trace inequalities from Hilbert space operators to a broader non-commutative setting.

Findings

Extended classical inequalities to $ au$-measurable operators

Established Clarkson inequalities for $n$-tuples of such operators

Presented a general parallelogram law for $ au$-measurable operators

Abstract

Let $\mathfrak{M}$ be a semifinite von Neumann algebra on a Hilbert space equipped with a faithful normal semifinite trace $\tau$. A closed densely defined operator $x$ affiliated with $\mathfrak{M}$ is called $\tau$-measurable if there exists a number $\lambda \geq 0$ such that $\tau \left(e^{|x|}(\lambda,\infty)\right)<\infty$. A number of useful inequalities, which are known for the trace on Hilbert space operators, are extended to trace on $\tau$-measurable operators. In particular, these inequalities imply Clarkson inequalities for $n$-tuples of $\tau$-measurable operators. A general parallelogram law for $\tau$-measurable operators are given as well.