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

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