On maps preserving zeros of the polynomial \(xy - yx^\ast\) (Q2568372)

In this interesting paper the authors apply a recently developed tool of functional identities to obtain the following result: Suppose \(\mathbb F\) is a field with char\(\mathbb F\not\in\{2,3\}\), and \(A=M_n(\mathbb F)\) is a matrix algebra with involution \(^\ast\). If \(\Theta:A\to A\) is a linear bijection such that \[ xy-yx^\ast=0\;\Longrightarrow\;\Theta(x)\Theta(y)-\Theta(y)\Theta(x)^\ast=0 \] then \(\Theta(x)=\lambda uxu^{-1}\), where \(\lambda\) and \(u^\ast u= uu^\ast\) are scalars. Here, the involution~\(^\ast\) is supposed to be merely additive, i.e., it satisfies (i) \((x+y)^\ast=x^\ast+y^\ast\), and (ii) \((xy)^\ast=y^\ast x^\ast\), and (iii) \(x^{\ast\ast}=x\). As such, they come in two flavors: the linear (e.g., transposition,\dots) and the conjugate-linear (e.g., Hilbert-space adjoint,\dots) ones. The purely algebraic arguments enables the authors to treat all involutions simultaneously. A negligible price to pay is that the matrices are assumed to be of dimension at least \(n\geq 20\). The paper starts with two open problems that are generalization of the above result, and continues with a brief survey of the theory of functional identities. Copious references are given to the historical background, the current importance of the skew-polynomial \(xy - yx^\ast\), as well as to the theory of functional identities.