On injective Jordan semi-triple maps of matrix algebras

We show that every injective Jordan semi-triple map on the algebra Mn(F)Mn(F) of all n × n   matrices with entries in a field FF (i.e. a map Φ:Mn(F)→Mn(F)Φ:Mn(F)→Mn(F) satisfyingΦ(ABA)=Φ(A)Φ(B)Φ(A)Φ(ABA)=Φ(A)Φ(B)Φ(A)for every A and B   in Mn(F)Mn(F)) is given by a map of the following form: there exist σ∈Fσ∈F, σ = ±1, an injective homomorphism ϕ   of FF and an invertible T∈Mn(F)T∈Mn(F) such that eitherΦ(A)=σT-1AϕTforallA∈Mn(F),orΦ(A)=σT-1AϕtTforallA∈Mn(F).Here, Aϕ is the image of A under ϕ applied entrywise.