Determinant preserving maps on matrix algebras (Q1601629)

Theorem: Let \(M_n=M_n(\mathbb C)\), and \(\varphi: M_n \rightarrow M_n\) be a not necessarily linear, surjective mapping satisfying \(\det(A+\lambda B)=\det(\varphi(A)+\lambda \varphi(B))\) for all \(A,B\in M_n\) and \(\lambda\in \mathbb C.\) Then for some \(M,N\in M_n\) with \(\det(MN)=1\) either \(\varphi(A)=MAN\) for all \(A\in M_n\) or \(\varphi(A)=MA^tN\) for all \(A\in M_n.\) The authors remark an analogy between their extension of Frobenius' 1897 theorem for linear determinant preservers and the extension by \textit{S. Kowalski} and \textit{Z. Slodkowski} [Stud. Math. 67, 215-223 (1980; Zbl 0456.46041)] of the Gleason-Kahane-Zelazko theorem for linear functionals on unital commutative complex Banach algebras.