Nonlinear anti-commuting maps of unital algebras with idempotents

Abstract

Let A be a unital algebra with a nontrivial idempotent e1. A map φ : A → A is anti-commuting if [φ(a), b] = −[a, φ(b)] holds for all a, b ∈ A. In this paper, we give a general form of φ on A; particularly, if A is prime, then such maps are either central-valued maps or of the forms a 7→ za+ f(a) for all a ∈ A, where z is in the center of A and f is a central-valued map. Mathematics Subject Classification: 47B47, 47B49, 46L10