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arXiv · 2402.13416

Birkhoff-James classification of norm's properties

Alexander Guterman·Bojan Kuzma·Sushil Singla·Svetlana Zhilina
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Abstract

For an arbitrary normed space $\mathcal X$ over a field $\mathbb F \in \{ \mathbb R, \mathbb C \}$, we define the directed graph $\Gamma(\mathcal X)$ induced by Birkhoff-James orthogonality on the projective space $\mathbb P(\mathcal X)$, and also its nonprojective counterpart $\Gamma_0(\mathcal X)$. We show that, in finite-dimensional normed spaces, $\Gamma(\mathcal X)$ 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 $\Gamma_0(\mathcal{R})$ of a (real or complex) Radon plane $\mathcal{R}$ is isomorphic to the graph $\Gamma_0(\mathbb F^2, \|\cdot\|_2)$ of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.

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Alexander Guterman, Bojan Kuzma, Sushil Singla, Svetlana Zhilina. 2024-02-20. Birkhoff-James classification of norm's properties. https://arxiv.org/abs/2402.13416

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