Dixmier traces are weak⁎ dense in the set of all fully symmetric traces

We extend Dixmier's construction of singular traces (see [2]) to arbitrary fully symmetric operator ideals. In fact, we show that the set of Dixmier traces is weak⁎ dense in the set of all fully symmetric traces (that is, those traces which respect Hardy–Littlewood submajorization). Our results complement and extend earlier work of Wodzicki [22].