On symmetry of the (strong) Birkhoff–James orthogonality in Hilbert C∗-modules

Abstract

In this note, we prove that the Birkhoff–James orthogonality, as well as the strong Birkhoff–James orthogonality, is a symmetric relation in a full Hilbert $\mathcal{A}$-module $V$ if and only if at least one of the underlying $C^{\ast }$-algebras $\mathcal{A}$ or $\mathbf{K}\left(V\right)$ is isomorphic to $\mathbb{C}$.