On bilinear maps determined by rank one idempotents
Using \(C^\ast\)-algebra techniques, the authors proved in [Stud. Math. 193, No. 2, 131--159 (2009; Zbl 1168.47029)] that for the algebra of square complex matrices and for a complex linear space \(X\), any bilinear map \(\phi:A\times A\rightarrow X\) is of the form \(\phi(a,b)=\phi_1(ab)+\phi_2(ba)\), where \(\phi_1,\phi_2:A\rightarrow X\) are linear maps, in case \(\phi(a,b)=0\) whenever \(ab=ba=0\). In this paper, based on elementary calculations, this result is extended to the algebra of all square matrices of a fixed order, over a field of characteristic different from \(2\), under the assumption \(\phi(e,f)=0\) whenever \(e\) and \(f\) are rank one idempotents that are orthogonal. Applications of this result to some linear preserver problems are presented.
scientific article; zbMATH DE number 5650007