Traces on symmetrically normed operator ideals

TL;DR

This paper characterizes when continuous singular traces exist on symmetrically normed operator ideals, compares their subclasses, and relates traces to symmetric functionals, extending results to semifinite von Neumann algebras.

Contribution

It provides new criteria for the existence of continuous singular traces, distinguishes subclasses respecting Hardy-Littlewood majorization, and establishes a bijection with symmetric functionals.

Findings

Criteria for existence of continuous singular traces.

The class of all continuous singular traces is broader than those respecting Hardy-Littlewood majorization.

A canonical bijection between traces and symmetric functionals is established.

Abstract

For every symmetrically normed ideal $\mathcal{E}$ of compact operators, we give a criterion for the existence of a continuous singular trace on $\mathcal{E}$. We also give a criterion for the existence of a continuous singular trace on $\mathcal{E}$ which respects Hardy-Littlewood majorization. We prove that the class of all continuous singular traces on $\mathcal{E}$ is strictly wider than the class of continuous singular traces which respect Hardy-Littlewood majorization. We establish a canonical bijection between the set of all traces on $\mathcal{E}$ and the set of all symmetric functionals on the corresponding sequence ideal. Similar results are also proved in the setting of semifinite von Neumann algebras.