Birkhoff-James classification of norm's properties

Abstract

For an arbitrary normed space $\mathcal{X}$ over a field $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, we define the directed graph $\hat{I}“(\mathcal{X})$ induced by Birkhoff-James orthogonality on the projective space $\mathbb{P}(\mathcal{X})$, and also its nonprojective counterpart $\hat{I}{“}_0(\mathcal{X})$. We show that, in finite-dimensional normed spaces, $\hat{I}“(\mathcal{X})$ carries all the information about the dimension, smooth points, and norm's maximal faces. It also allows to determine whether the norm is a supremum norm or not, and thus classifies finite-dimensional abelian $C^{\ast }$-algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph $\hat{I}{“}_0(\mathcal{R})$ of a (real or complex) Radon plane $\mathcal{R}$ is isomorphic to the graph $\hat{I}{“}_0({\mathbb{F}}^2,\Vert \cdot {\Vert }_2)$ of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.

Accepted for publications in AOT in The Special Issue Dedicated to Professor Chi-Kwong Li