Achievement of continuity of $(\varphi ,\psi )$-derivations without continuity

Abstract

Suppose that $\calak$ is a $C^{\ast }$-algebra acting on a Hilbert space $\calhk$, and that $\varphi ,\psi$ are mappings from $\calak$ into $B(\calhk)$ which are not assumed to be necessarily linear or continuous. A $(\varphi ,\psi )$-derivation is a linear mapping $d: \calak \to B(\calhk)$ such that $$d(ab)=ϕ(a)d(b)+d(a)ψ(b)\quad (a,b\in \calak).$$ We prove that if $\varphi$ is a multiplicative (not necessarily linear) $\ast$-mapping, then every $\ast$-$(\varphi ,\varphi )$-derivation is automatically continuous. Using this fact, we show that every $\ast$-$(\varphi ,\psi )$-derivation $d$ from $\calak$ into $B(\calhk)$ is continuous if and only if the $\ast$-mappings $\varphi$ and $\psi$ are left and right $d$-continuous, respectively.

To appear in Bull. Belgian Math Soc