Automatic continuity of $\sigma$-derivations on $C^*$-algebras

Abstract:

Let $\mathcal {A}$ be a $C^*$-algebra acting on a Hilbert space $\mathcal {H}$, let $\sigma :\mathcal {A}\to B(\mathcal {H})$ be a linear mapping and let $d:\mathcal {A}\to B(\mathcal {H})$ be a $\sigma$-derivation. Generalizing the celebrated theorem of Sakai, we prove that if $\sigma$ is a continuous $*$-mapping, then $d$ is automatically continuous. In addition, we show the converse is true in the sense that if $d$ is a continuous $*$-$\sigma$-derivation, then there exists a continuous linear mapping $\Sigma :\mathcal {A}\to B(\mathcal {H})$ such that $d$ is a $*$-$\Sigma$-derivation. The continuity of the so-called $*$-$(\sigma ,\tau )$-derivations is also discussed.