Singular Traces and Compact Operators. I

We give a necessary and sufficient condition on a positive compact operator $T$ for the existence of a singular trace (i.e. a trace vanishing on the finite rank operators) which takes a finite non-zero value on $T$. This generalizes previous results by Dixmier and Varga. We also give an explicit description of these traces and associated ergodic states on ${\mathrm{\ell }}^{\mathrm{\infty }}(\mathbb{N})$ using tools of non standard analysis in an essential way.