Dixmier traces are weak$^*$ dense in the set of all fully symmetric traces

TL;DR

This paper demonstrates that Dixmier traces are weak$^*$ dense within all fully symmetric traces on operator ideals, extending their construction and connecting to Hardy-Littlewood submajorization.

Contribution

It extends Dixmier's construction of singular traces to all fully symmetric operator ideals and proves their weak$^*$ density among all such traces.

Findings

Dixmier traces are weak$^*$ dense in fully symmetric traces.

Extension of Dixmier's construction to arbitrary fully symmetric ideals.

Results complement and extend Wodzicki's earlier work.

Abstract

We extend Dixmier's construction of singular traces (see \cite{Dixmier}) to arbitrary fully symmetric operator ideals. In fact, we show that the set of Dixmier traces is weak$^*$ dense in the set of all fully symmetric traces (that is, those traces which respect Hardy-Littlewood submajorization). Our results complement and extend earlier work of Wodzicki \cite{Wodzicki}.