Orthogonal forms and orthogonality preservers on real function algebras revisited

Abstract

In 2014, we determine the precise form of a continuous orthogonal form on a commutative real C$^*$-algebra. We also describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between commutative unital real C$^*$-algebras. Among the consequences, we show that every orthogonality preserving linear bijection between commutative unital real C$^*$-algebras is continuous. In this note we revisit these results and their proofs with the idea of filling a gap in the arguments, and to extend the original conclusions.