From norm derivatives to orthogonalities in Hilbert $C^{\ast }$-modules

Abstract

Let $(\mathcal{X},⟨\cdot ,\cdot ⟩)$ be a Hilbert $C^{\ast }$-module over a $C^{\ast }$-algebra $\mathcal{A}$ and let $\mathcal{S}(\mathcal{A})$ be the set of states on $\mathcal{A}$. In this paper, we first compute the norm derivative for elements $x$ and $y$ of $\mathcal{X}$ as follows \begin{align*} ρ_{_{+}}(x, y) = \max\Big\{\mbox{Re}\,φ(\langle x, y\rangle): \, φ\in \mathcal{S}(\mathscr{A}), φ(\langle x, x\rangle) = \|x\|^2\Big\}. \end{align*} We then apply it to characterize different concepts of orthogonality in $\mathcal{X}$. In particular, we present a simpler proof of the classical characterization of Birkhoff--James orthogonality in Hilbert $C^{\ast }$-modules. Moreover, some generalized Daugavet equation in the $C^{\ast }$-algebra $\mathbb{B}(\mathcal{H})$ of all bounded linear operators acting on a Hilbert space $\mathcal{H}$ is solved.

13 pages

References in corpus (1)

• An extension of orthogonality relations based on norm derivatives