Commutativity preserving linear maps on central simple algebras. (Q1764832)

Let \(A\) be a unital algebra over a field \(F\) and \(\varphi\colon A\to A\) a linear map. Then \(\varphi\) is called commutativity-preserving if for all \(x,y\in A\) such that \(xy=yx\), \(\varphi(x)\varphi(y)=\varphi(y)\varphi(x)\). Among the obvious examples are the standard commutativity-preserving maps, defined by \(\varphi(x)=\lambda\psi(x)+\mu(x)1\), where \(\lambda\in F\), \(\psi\) is an automorphism or antiautomorphism of \(A\), and \(\mu\) is a linear functional on \(A\). In this paper it is shown that if \(\text{char}(F)=0\) and \(A\) is a central simple algebra with \(4\neq\dim_FA<\infty\), then a linear map \(\varphi\) such that \(\varphi(x^2)\varphi(x)=\varphi(x)\varphi(x^2)\) for all \(x\in A\) is either a standard commutativity-preserving map or a map with commutative range. This result extends a recent result of \textit{M. Omladič}, \textit{H. Radjavi} and \textit{P. Šemrl} [J. Pure Appl. Algebra 156, No. 2-3, 309-328 (2001; Zbl 0973.15002)], which dealt with the case of an algebraically closed \(F\) of characteristic \(0\) and \(A=M_n(F)\), \(n\neq 2\).