Abstract. For $A\in{\cal L}(H)$ (the algebra of all operators on the separable complex Hilbert space $H$) we study the property of the range of $\delta_A(X\mapstochar\rightarrow AX-XA)$ if, for $T\in{\cal C}_1(H)$ (trace class operators) $AT=TA$ implies $A^*T=TA^*$. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of $\delta_A$ is closed under taking adjoints.

AMS Subject Classification (1991): 47B47

Reçu le 12 février, 1993. (Registered under 5565/2009.)