Inequalities for trace on $tau$-measurable operators

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.