Automorphisms and derivations of finite-dimensional algebras

Let A be a finite-dimensional algebra over a field F with char(F) not equal 2. We show that a linear map D : A -> A satisfying xD(x)x is an element of[A, A] for every x is an element of A is the sum of an inner derivation and a linear map whose image lies in the radical of A. Assuming additionally that A is semisimple and char(F) not equal 3, we show that a linear map T : A -> A satisfies T(x)(3) -x(3) is an element of[A, A] for every x is an element of A if and only if there exist a Jordan automorphism J of A lying in the multiplication algebra of A and a central element alpha satisfying alpha(3) = 1 such that T(x) = alpha J(x) for all x is an element of A. These two results are applied to the study of local derivations and local (Jordan) automorphisms. In particular, the second result is used to prove that every local Jordan automorphism of a finite-dimensional simple algebra A (over a field F with char(F) not equal 2, 3) is a Jordan automorphism. (c) 2022 Elsevier Inc. All rights reserved.