A Characterization of $(\sigma ,\tau )-$ 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 $\sigma ,\tau$ on $\mathcal{A}$, every bounded linear map $f:\mathcal{A}\to \mathcal{M}$ with property that $f(p^2)=\sigma (p)f(p)+f(p)\tau (p)$ for every projection $p$ in $\mathcal{A}$ is a $(\sigma ,\tau )-$derivation. Also, it is shown that a bounded linear map $f:\mathcal{A}\to \mathcal{M}$ which satisfies $f(ab)=\sigma (a)f(b)+f(a)\tau (b)$ for all $a,b\in \mathcal{A}$ with $ab=S$, is a $(\sigma ,\tau )-$ derivation if $\tau (S)$ is left invertible for fixed $S$.