2016/12/14 by Semenov, Evgenii, Sukochev, Fedor, Usachev, Aleksandr +1
#FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1612.04509
We introduce a new approach to traces on the principal ideal \mathcal L1,∞ 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 L1,∞ 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 L1,∞ and the set of all translation invariant functionals on l_∞. 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_∞, and definitively classify the measurability of operators in \mathcal L1,∞ 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 L1,∞) from~\citeCS. As an application we extend Connes' classical trace theorem to positive normalised traces.