  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.
