Communications of the
Korean Mathematical Society CKMS

In this paper, we introduce some notions of orthogonality in the setting of Finsler $C^*$-modules and investigate their relations with the Birkhoff-James orthogonality. Suppose that $(E, \rho )$ and $(F, \rho')$ are Finsler modules over $C^*$-algebras $\mathcal{A}$ and $\mathcal{B}$, respectively, and $\varphi: \mathcal{A}\to \mathcal{B}$ is a $*$-homomorphism. A map $\Psi: E\to F$ is said to be a $\varphi$-morphism of Finsler modules if $\rho' (\Psi(x))=\varphi(\rho (x))$ and $\Psi(ax)=\varphi(a)\Psi(x)$ for all $a\in \mathcal{A}$ and all $x\in E$. We show that each $\varphi$- morphism of Finsler $C^*$-modules preserves the Birkhoff-James orthogonality and conversely, each surjective linear map between Finsler $C^*$-modules preserving the Birkhoff-James orthogonality is a $\varphi$-morphism under certain conditions. In fact, we state a version of Wigner's theorem in the framework of Finsler $C^*$-modules.

Keywords: Finsler $C^*$-modules, orthogonality, $\varphi$-morphism of Finsler $C^*$-modules

MSC numbers: Primary 46L08; Secondary 46L05, 47B65