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We propose a new class of traces motivated by a trace/trace class propertydiscovered by Laurie, Nordgren, Radjavi and Rosenthal concerning products ofoperators outside the trace class. Spectral traces, traces that depend only onthe spectrum and algebraic multiplicities, possess this property and we suspectothers do, but we know of no other traces that do.  This paper is intended to be part survey. We provide here a brief overview ofsome facts concerning traces on ideals, especially involving Lidskii formulasand spectral traces.  We pose the central question: whenever the relevant products, $AB$, $BA$ liein an ideal, do bounded operators $A$, $B$ always commute under any trace onthat ideal, i.e.,$\tau(AB) = \tau(BA)$? And if not, characterize whichtraces/ideals do possess this property.

Traces on ideals and the commutator property