Derivations and local derivations on Freudenthal almost f-algebras

Let A be an Archimedean f  -algebra and let N(A)N(A) be the set of all nilpotent elements of A. Colville et al. [4] proved that a positive linear map d:A→Ad:A→A is a derivation if and only if d(A)⊂N(A)d(A)⊂N(A) and d(A2)={0}d(A2)={0}, where A2A2 is the set of all products ab in A.In this paper, we establish a result corresponding to the Colville–Davis–Keimel theorem for arbitrary derivation d on Freudenthal almost f-algebras. Moreover, we prove that any local derivation on a Freudenthal almost f-algebra A  , such that N(A)={a∈A;a2=0}N(A)={a∈A;a2=0}, is a derivation.