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

Abstract

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