Automatic Continuity of $\sigma$-Derivations on $C^{\ast }$-Algebras

Abstract

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

8 pages, some minor corrections, to appear in Proc. Amer. Math. Soc

References in corpus (1)

• Deformations of Lie Algebras using $\sigma$-derivations