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

Abstract

Let {{mathcal {M}}} be a semifinite von Neumann algebra with a faithful normal semifinite trace tau and let {{mathcal {A}}} be an arbitrary von Neumann subalgebra of {{mathcal {M}}}. We characterize the class of symmetric ideals {{mathcal {E}}} in {{mathcal {M}}} such that derivations delta :{{mathcal {A}}}rightarrow {{mathcal {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 {{mathcal {A}}} into the ideal {{mathcal {C}}}_0({{mathcal {M}}},tau ) of all tau -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).