Nonlinear strong commutativity preserving maps on skew elements of prime rings with involution

Abstract

Let A be a prime ring of characteristic not 2, with involution, with center Z(A) and with skew elements K. Suppose that f:K→A is a map satisfying [f(x),f(y)]=[x,y] for all x,y∈K. Then there exists a map μ:K→Z(A) such that f(x)=x+μ(x) for all x∈K or f(x)=-x+μ(x) for all x∈K except when A is an order in a 4, 9 or 16-dimensional central simple algebra.