Banach limits and traces on $\mathcal L_{1,\infty}$

TL;DR

This paper introduces a new explicit method to construct and classify traces on the ideal L_{1,} using translation invariant functionals, resolving open problems and extending Connes' trace theorem.

Contribution

It provides a novel bijection between traces on L_{1,} and translation invariant functionals on l_, offering a comprehensive classification and new insights.

Findings

Classified measurability of operators in L_{1,} based on eigenvalue sums.

Extended Connes' trace theorem to positive normalized traces.

Resolved several open problems related to traces on L_{1,}.

Abstract

We introduce a new approach to traces on the principal ideal $\mathcal L_{1,\infty}$ generated by any positive compact operator whose singular value sequence is the harmonic sequence. Distinct from the well-known construction of J.~Dixmier, the new approach provides the explicit construction of every trace of every operator in $\mathcal L_{1,\infty}$ in terms of translation invariant functionals applied to a sequence of restricted sums of eigenvalues. The approach is based on a remarkable bijection between the set of all traces on $\mathcal L_{1,\infty}$ and the set of all translation invariant functionals on $l_\infty$. This bijection allows us to identify all known and commonly used subsets of traces (Dixmier traces, Connes-Dixmier traces, etc.) in terms of invariance properties of linear functionals on $l_\infty$, and definitively classify the measurability of operators in $\mathcal L_{1,\infty}$ in terms of qualified convergence of sums of eigenvalues. This classification has led us to a resolution of several open problems (for the class $\mathcal L_{1,\infty}$) from~\cite{CS}. As an application we extend Connes' classical trace theorem to positive normalised traces.