Singular Traces and Compact Operators

We give a necessary and sufficient condition on a positive compact operatorTfor the existence of a singular trace (i.e. a trace vanishing on the finite rank operators) which takes a finite non-zero value onT. This generalizes previous results by Dixmier and Varga. We also give an explicit description of these traces and associated ergodic states onl∞(N) using tools of non standard analysis in an essential way.