Derivations on ternary rings of operators

To each projection $p$ in a $C^*$-algebra $A$ we associate a family of derivations on $A$, called $p$-derivations, and relate them to the space of triple derivations on $p A (1-p)$. We then show that every derivation on a ternary ring of operators is spatial and we investigate whether every such derivation on a weakly closed ternary ring of operators is inner.