Jordan and Jordan Higher All-derivable Points of Some Algebras

In this paper, we characterize Jordan derivable mappings in terms of Peirce decomposition and determine Jordan all-derivable points for some general bimodules. Then we generalize the results to the case of Jordan higher derivable mappings. An immediate application of our main results shows that for a nest $\mathcal{N}$ on a Banach $X$ with the associated nest algebra $alg\mathcal{N}$, if there exists a non-trivial element in $\mathcal{N}$ which is complemented in $X$, then every $C\in alg\mathcal{N}$ is a Jordan all-derivable point of $L(alg\mathcal{N}, B(X))$ and a Jordan higher all-derivable point of $L(alg\mathcal{N})$.