## Abstract

For an arbitrary normed space X over a field F∈{R,C}, we define the directed graph Γ(X) induced by Birkhoff–James orthogonality on the projective space P(X), and also its nonprojective counterpart Γ_{0}(X). We show that, in finite-dimensional normed spaces, Γ(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^{∗}-algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph Γ_{0}(R) of a (real or complex) Radon plane R is isomorphic to the graph Γ_{0}(F^{2},‖·‖_{2}) of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.

Original language | English |
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Article number | 43 |

Journal | Advances in Operator Theory |

Volume | 9 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2024 |

### Bibliographical note

Publisher Copyright:© The Author(s) 2024.

## Keywords

- 05C63
- 46B20
- Birkhoff–James orthogonality
- Dimension
- Normed space
- Orthodigraph
- Radon plane
- Rotundness
- Smoothness