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].