Traces on symmetrically normed operator ideals

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.

submitted to Crelle

References in corpus (2)

• Dimensions and singular traces for spectral triples, with applications to fractals
• Dixmier Traces as Singular Symmetric Functionals and Applications to Measurable Operators