Orthogonal forms and orthogonality preservers on real function algebras

Abstract

We initiate the study of orthogonal forms on a real C${}^{\ast }$-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${}^{\ast }$-algebra, $A$, there exist functionals ${\phi }_1$ and ${\phi }_2$ in $A^{\ast }$ satisfying $$V(x,y)={\phi }_1(xy)+{\phi }_2(x{y}^{\ast }),$$ 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${}^{\ast }$-algebras. As a consequence, we show that every orthogonality preserving linear bijection between unital commutative real C${}^{\ast }$-algebras is continuous.

To appear in Linear and Multilinear Algebra