Jordan (α,β)-derivations on triangular algebras and related mappings

Let R be a 2-torsion free commutative ring with identity, A,B be unital algebras over R and M be a unital (A,B)-bimodule, which is faithful as a left A-module and also as a right B-module. Let T=AM0B be the triangular algebra consisting of A,B and M, and let d be an R-linear mapping from T into itself. Suppose that A and B have only trivial idempotents. Then the following statements are equivalent: (1) d is a Jordan (α,β)-derivation on T; (2) d is a Jordan triple (α,β)-derivation on T; (3) d is an (α,β)-derivation on T. Furthermore, a generalized version of this result is also given. We characterize the actions of automorphisms and skew derivations on the triangular algebra T. The structure of continuous (α,β)-derivations of triangular Banach algebras and that of generalized Jordan (α,β)-derivations of upper triangular matrix algebras are described.