A Characterization of $(σ,τ)-$ derivations on von Neumann algebras

Abstract

Let $\mathcal A$ be a von Neumann algebra and $\mathcal M$ be a Banach $\mathcal A-$module. It is shown that for every homomorphisms $σ, τ$ on $\mathcal A$, every bounded linear map $f:\mathcal A\to \mathcal M$ with property that $f(p^2)=σ(p)f(p)+f(p)τ(p)$ for every projection $p$ in $\mathcal A$ is a $(σ,τ)-$derivation. Also, it is shown that a bounded linear map $f:\mathcal A \to \mathcal M $ which satisfies $f(ab)= σ(a)f(b)+f(a)τ(b)$ for all $a,b\in \mathcal A$ with $ab=S$, is a $(σ,τ)-$ derivation if $τ(S)$ is left invertible for fixed $S$.