Orthogonal forms and orthogonality preservers on real function algebras

Abstract

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 on a commutative real C-algebra,, there exist functionals and in satisfying for every in. 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.