Characterization of singular traces on the weak trace class ideal generated by exponentiation invariant extended limits

This paper studies the subset of singular traces generated by exponentiation-invariant extended limits. We describe relations between this subset and other important subsets of singular traces. We prove several conditions for measurability of operators from the weak trace class ideal with respect to the traces generated by exponentiation-invariant extended limits. We resolve an open question raised in [S. Lord, F. Sukochev, Measure theory in noncommutative spaces, SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), paper 072, 36] in the setting of the weak trace class ideal.