Derivations in the Banach ideals of $\tau$-compact operators

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semi-finite trace $\tau$ and let $S_0(\tau )$ be the algebra of all $\tau$-compact operators affiliated with $\mathcal{M}$. Let $E(\tau)\subseteq S_0(\tau)$beasymmetricoperatorspace(on$$\mathcal{M}$$)andlet$$\mathcal{E}$$beasymmetrically-normedBanachidealof$\tau$-compact operators in $\mathcal{M}$. We study (i) derivations $\delta$ on $\mathcal{M}$ with the range in $E(\tau )$ and (ii) derivations on the Banach algebra $\mathcal{E}$. 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 $E(\tau )$). In the second case we show that any such derivation is necessarily inner when $\mathcal{M}$ is a type $I$ factor. As an interesting application of our results for the case (i) we deduce that any derivation from $\mathcal{M}$ into an $L_p$-space, $L_p(\mathcal{M},\tau )$, ($1<p<\mathrm{\infty }$) associated with $\mathcal{M}$ is inner.