Orthogonal forms and orthogonality preservers on real function algebras

We initiate the study of orthogonal forms on a real C$^*$-algebra. Motivated by previous contributions, due to Ylinen, Jajte, Paszkiewicz and Goldstein, we prove that for every continuous orthogonal form $V$ on a commutative real C$^*$-algebra, $A$, there exist functionals $\varphi_1$ and $\varphi_2$ in $A^{*}$ satisfying $$V(x,y) = \varphi_1 (x y) + \varphi_2 (x y^*),$$ for every $x,y$ in $A$. We describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between unital commutative real C$^*$-algebras. As a consequence, we show that every orthogonality preserving linear bijection between unital commutative real C$^*$-algebras is continuous.Comment: To appear in Linear and Multilinear Algebr