Additive maps onto matrix spaces compressing the spectrum

We prove that given a unital C⁎C⁎-algebra AA and an additive and surjective map T:A→MnT:A→Mn such that the spectrum of T(x)T(x) is a subset of the spectrum of x   for each x∈Ax∈A, then T   is either an algebra morphism, or an algebra anti-morphism. We arrive at the same conclusion for an arbitrary unital, complex Banach algebra AA, by imposing an extra surjectivity condition on the map T.