Maps on spaces of linear transformations over semisimple algebras
It is well-known that for any field K every bijective linear map f on \(M_ n(K)\) which preserves the determinant function is an equivalence transformation \((f(A)=PAQ\) for some invertible matrices P and Q) or the composite of an equivalence transformation with the transpose transformation \((f(A)=A^ T)\). A theorem of \textit{J. Dieudonné} [Arch. Math. 1, 282-287 (1949; Zbl 0032.10601)] shows that, in fact, this is true for the linear map which preserves the set of singular matrices, and \textit{M. Marcus} and \textit{B. Moyls} [Pac. J. Math. 9, 1215-1221 (1959; Zbl 0089.089)] showed that the same holds for the bijective linear maps which preserve the set of matrices of rank 1. In this paper the author extends the problem to maps on spaces of linear transformations over semisimple algebras, preserving transformations of local length 1, a notion defined by the author as an extension of the property of having rank 1. He also generalizes Dieudonné's theorem by determining the maps preserving the set of ``locally singular'' transformations.
linear transformations over semisimple algebras
scientific article; zbMATH DE number 4075317