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Journal of Operator Theory

Volume 89, Issue 2, Spring 2023  pp. 305-342.

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

Authors:  Galina Levitina (1), Alexandr Usachev (2)
Author institution: (1) Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
(2) School of Mathematics and Statistics, Central South University, 410085, Hunan, China

Summary:  In this paper we extend the Pietsch correspondence for ideals of compact operators and traces on them to the semifinite setting. We establish a correspondence between a shift monotone space $E(\mathbb Z)$ of sequences and a Calkin space $E(\mathcal M,\tau)$ of $\tau$-measurable operators affiliated with a semifinite von Neumann algebra $\mathcal M$ equipped with a faithful normal semifinite trace $\tau$ as well as a correspondence between shift invariant functionals on $E(\mathbb Z)$ and symmetric functionals on $E(\mathcal M,\tau)$.

Keywords:  symmetric functionals, singular traces, von Neumann algebras, Calkin spaces, shift monotone spaces

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