Orthogonal forms and orthogonality preservers on real function algebras revisited

Abstract

In [Garcés J, Peralta AM. Orthogonal forms and orthogonality preservers on real function algebras. Linear Multilinear Algebra. 2014;62:275–296.] 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 aim of filling a gap in the arguments, and to extend the original conclusions.