On injective Jordan semi-triple maps of matrix algebras

We show that every injective Jordan semi-triple map on the algebra of all n × n matrices with entries in a field (i.e. a map satisfyingΦ(ABA)=Φ(A)Φ(B)Φ(A)for every A and B in ) is given by a map of the following form: there exist , σ = ±1, an injective homomorphism of and an invertible such that either Here, A is the image of A under applied entrywise.