Maps on matrix algebras preserving idempotents

AbstractLet Mn be the algebra of all n×n complex matrices and Pn the set of all idempotents in Mn. Suppose φ:Mn→Mn is a surjective map satisfying A−λB∈Pn if and only if φ(A)−λφ(B)∈Pn, A,B∈Mn, λ∈C. Then either φ is of the form φ(A)=TAT−1, A∈Mn, or φ is of the form φ(A)=TAtT−1, A∈Mn, where T∈Mn is a nonsingular matrix