  In this paper we extend the Pietsch correspondence for ideals of compact
operators and traces on them to the semifinite setting. We prove that a
shift-monotone space $E(\Z)$ of sequences indexed by $\Z$ defines a Calkin
space $E(\cM,\tau)$ of $\tau$-measurable operators affiliated with a semifinite
von Neumann algebra $\cM$ equipped with a faithful normal semifinite trace
$\tau$. Furthermore, we show that shift-invariant functionals on $E(\Z)$
generate symmetric functionals on $E(\cM,\tau)$. In the special case, when the
algebra $\cM$ is atomless or atomic with atoms of equal trace, the converse
also holds and we have a bijective correspondence between all shift-monotone
spaces $E(\Z)$ and Calkin spaces $E(\cM,\tau)$ as well as a bijective
correspondence between shift-invariant functionals on $E(\Z)$ and symmetric
functionals on $E(\cM,\tau)$. The bijective correspondence
$E(\Z)\leftrightarrows E(\cM,\tau)$ extends to a correspondence between
complete symmetrically $\Delta$-normed spaces $E(\cM,\tau)$ and complete
$\Delta$-normed shift-monotone spaces $E(\Z)$.
