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

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.

accepted to Advances in Mathematics

References in corpus (2)

• Traces of compact operators and the noncommutative residue
• Measure Theory in Noncommutative Spaces