Isolated vertices and diameter of the $BJ$-orthograph in $C^*$-algebras

TL;DR

This paper characterizes isolated vertices in the BJ-orthograph of $C^*$-algebras and shows that, except for a few cases, the non-isolated vertices form a connected component with diameter at most 4.

Contribution

It provides necessary and sufficient conditions for isolated vertices and establishes the diameter bound of the non-isolated component in the BJ-orthograph of $C^*$-algebras.

Findings

Characterization of isolated vertices in the orthograph.

Non-isolated points form a connected component with diameter ≤ 4.

Exceptions include $ ext{C}$, $ ext{C} igoplus ext{C}$, and $M_2( ext{C})$.

Abstract

We give necessary and sufficient condition that an element of an arbitrary $C^{*}$-algebra is an isolated vertex of the orthograph related to the mutual strong Birkhoff-James orthogonality. Also, we prove that for all $C^{*}$-algebras except $\mathbb{C},\mathbb{C}\oplus\mathbb{C}$ and $M_2(\mathbb{C})$ all non isolated points make a single connected component of the orthograph which diameter is less than or equal to $4$, i.e. any two non isolated points can be connected by a path with at most $4$ edges. Some related results are given.