Pietsch correspondence for symmetric functionals on Calkin operator spaces associated with semifinite von Neumann algebras

Abstract: 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 $E(\Z)$.