Derivations with Values in the Ideal of $$\tau $$-Compact Operators Affiliated with a Semifinite von Neumann Algebra

Abstract Let $${{\mathcal {M}}}$$ M be a semifinite von Neumann algebra with faithful normal trace $$\tau $$ ? and let {A}}}$$ A an arbitrary subalgebra of . We characterize the class symmetric ideals {E}}}$$ E in such that derivations $$\delta :{{\mathcal {A}}}\rightarrow {{\mathcal ? : ? are necessarily inner, which is unification far-reaching extension results due to Johnson Parrott (J Funct Anal 11:39–61, 1972), Kaftal Weiss 62:202–220, 1985), Popa 71:393–408, 1987). In particular, we show every derivation from into ideal {C}}}_0({{\mathcal {M}}},\tau )$$ C 0 ( , ) all -compact operators establishing version Johnson–Parrott–Popa Theorem different R?dulescu (Duke Math J 57(2):485–518, 1988, 1.1) contrasts example non-inner established (1988, 1.2).