Orthogonal forms and orthogonality preservers on real function algebras revisited

TL;DR

This paper revisits and extends previous results on the structure of orthogonal forms and orthogonality-preserving maps in commutative real C*-algebras, clarifying proofs and broadening applicability.

Contribution

It provides a refined analysis of orthogonal forms and maps, filling gaps in earlier proofs and extending the results to more general settings.

Findings

Every orthogonality preserving linear bijection between commutative unital real C*-algebras is continuous.

The form of continuous orthogonal forms on commutative real C*-algebras is precisely characterized.

General forms of orthogonality-preserving linear maps are described, including non-continuous cases.

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.