Let M be a von Neumann algebra equipped with a faithful normal
semi-finite trace Ο and let S0β(Ο) be the algebra of all
Ο-compact operators affiliated with M. Let E(Ο)βS0β(Ο) be a symmetric operator space (on M) and let
E be a symmetrically-normed Banach ideal of Ο-compact
operators in M. We study (i) derivations Ξ΄ on M
with the range in E(Ο) and (ii) derivations on the Banach algebra
E. In the first case our main results assert that such derivations
are continuous (with respect to the norm topologies) and also inner (under some
mild assumptions on E(Ο)). In the second case we show that any such
derivation is necessarily inner when M is a type I factor. As an
interesting application of our results for the case (i) we deduce that any
derivation from M into an Lpβ-space, Lpβ(M,Ο),
(1<p<β) associated with M is inner