Characterization of some derivations on von Neumann algebras via left centralizers

Abstract

Let $\mathfrak {M}$ be a von Neumann algebra, and let $\mathfrak {T}:\mathfrak {M} \rightarrow \mathfrak {M}$ be a bounded linear map satisfying $\mathfrak {T}(P^{2}) = \mathfrak {T}(P)P + \Psi (P,P)$ for each projection P of $\mathfrak {M}$, where $\Psi :\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}$ is a bi-linear map. If $\Psi$ is a bounded l-semi Hochschild 2-cocycle, then $\mathfrak {T}$ is a left centralizer associated with $\Psi$. By applying this conclusion, we offer a characterization of left $\sigma$-centralizers, generalized derivations and generalized $\sigma$-derivations on von Neumann algebras. Moreover, it is proved that if $\mathfrak {M}$ is a commutative von Neumann algebra and $\sigma :\mathfrak {M} \rightarrow \mathfrak {M}$ is an endomorphism, then every bi-$\sigma$-derivation $D:\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}$ is identically zero.