Let $M_n$ denote the algebra of $n \times n$ complex matrices and let $\mathcal{ A}\subseteq M_n$ be an arbitrary structural matrix algebra, i.e. a subalgebra of $M_n$ that contains all diagonal matrices. We consider injective maps $\phi : \mathcal{ A}\to M_n$ that satisfy the condition $$ \phi(X \bullet Y) = \phi(X) \bullet \phi(Y), \quad \text{ for all } X,Y \in \mathcal{ A}, $$ where $\bullet$ is either the standard matrix multiplication $(X,Y)\mapsto XY$, the Jordan product $(X,Y) \mapsto XY+YX$, or the normalized Jordan product $(X,Y) \mapsto \frac{ 1}{ 2}(XY+YX)$. We show that all such maps $\phi$ are automatically additive if and only if $\mathcal{ A}$ does not contain a central rank-one idempotent. Moreover, in this case, we fully characterize the form of these maps.
Categories: math.RA