Inequalities for trace on $τ$-measurable operators

Abstract

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