  This paper formulates Young-type inequalities for singular values (or
$s$-numbers) and traces in the context of von Neumann algebras. In particular,
it shown that if $\t(\cdot)$ is a faithful semifinite normal trace on a
semifinite von Neumann algebra $M$ and if $p$ and $q$ are positive real numbers
for which $p^{-1}+q^{-1}=1$, then, for all positive operators $a,b\in M$,
$\t(|ab|)\le p^{-1}\t(a^p)+ q^{-1}\t(b^q)$, with equality holding (in the cases
where $p^{-1}\t(a^p)+ q^{-1}\t(b^q)<\infty$) if and only if $b^q=a^p$.
