Idempotence-preserving maps without the linearity and surjectivity assumptions

AbstractLet Mn(F) be the space of all n×n matrices over a field F of characteristic not 2, and let Pn(F) be the subset of Mn(F) consisting of all n×n idempotent matrices. We denote by Φn(F) the set of all maps from Mn(F) to itself satisfying A−λB∈Pn(F) if and only if φ(A)−λφ(B)∈Pn(F) for every A,B∈Mn(F) and λ∈F. It was shown that φ∈Φn(F) if and only if there exists an invertible matrix P∈Mn(F) such that either φ(A)=PAP−1 for every A∈Mn(F), or φ(A)=PATP−1 for every A∈Mn(F). This improved Dolinar's result by omitting the surjectivity assumption and extending the complex field to any field of characteristic not 2