Achievement of continuity of $(ϕ,ψ)$-derivations without continuity

Abstract

Suppose that $\calak$ is a $C^*$-algebra acting on a Hilbert space $\calhk$, and that $ϕ, ψ$ are mappings from $\calak$ into $B(\calhk)$ which are not assumed to be necessarily linear or continuous. A $(ϕ, ψ)$-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 $ϕ$ is a multiplicative (not necessarily linear) $*$-mapping, then every $*$-$(ϕ,ϕ)$-derivation is automatically continuous. Using this fact, we show that every $*$-$(ϕ,ψ)$-derivation $d$ from $\calak$ into $B(\calhk)$ is continuous if and only if the $*$-mappings $ϕ$ and $ψ$ are left and right $d$-continuous, respectively.