Functional identities on Banach algebras and matrix rings

Abstract

Let A be a unital Banach algebra and M be a unital A -bimodule. We show that additive mappings δ , τ : A → M satisfy the identity A − 1 δ ( A ) + δ ( A ) A − 1 + A τ ( A − 1 ) + τ ( A − 1 ) A = 0 for all invertible element A in A if and only if there exists a Jordan derivation D such that δ ( A ) = D ( A ) + A δ ( I ) and τ ( A ) = D ( A ) − δ ( I ) A for all A ∈ A . We also characterize additive mappings δ , τ : A → M satisfying the identity δ ( A ) B + A τ ( B ) = M for all A , B ∈ A with AB = N, where N is an invertible element in A and M is a fixed element in M . Similar conclusions are obtained for additive mappings from a matrix ring M n ( D ) into its unital bimodule satisfying any of the above identities, where D is a division ring and char D ≠ 2 .