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

Abstract

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