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We introduce a new approach to traces on the principal ideal $\mathcalL_{1,\infty}$ generated by any positive compact operator whose singular valuesequence is the harmonic sequence. Distinct from the well-known construction ofJ.~Dixmier, the new approach provides the explicit construction of every traceof every operator in $\mathcal L_{1,\infty}$ in terms of translation invariantfunctionals applied to a sequence of restricted sums of eigenvalues. Theapproach 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 usedsubsets of traces (Dixmier traces, Connes-Dixmier traces, etc.) in terms ofinvariance properties of linear functionals on $l_\infty$, and definitivelyclassify the measurability of operators in $\mathcal L_{1,\infty}$ in terms ofqualified convergence of sums of eigenvalues. This classification has led us toa resolution of several open problems (for the class $\mathcal L_{1,\infty}$)from~\cite{CS}. As an application we extend Connes' classical trace theorem topositive normalised traces.

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