Additive maps onto matrix spaces compressing the spectrum

We prove that given a unital C*-algebra A and an additive and surjective map T : A -> M-n, such that the spectrum of T(x) is a subset of the spectrum of x for each x is an element of 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 A, by imposing an extra surjectivity condition on the map T. (C) 2016 Elsevier Inc. All rights reserved.