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

Abstract

AbstractLet $${{\mathcal {M}}}$$ M be a semifinite von Neumann algebra with a faithful normal semifinite trace $$\tau $$ τ and let $${{\mathcal {A}}}$$ A be an arbitrary von Neumann subalgebra of $${{\mathcal {M}}}$$ M . We characterize the class of symmetric ideals $${{\mathcal {E}}}$$ E in $${{\mathcal {M}}}$$ M such that derivations $$\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {E}}}$$ 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 $${{\mathcal {A}}}$$ A into the ideal $${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$$ C 0 ( M , τ ) of all $$\tau $$ 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).