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

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 C,C⊕C and M2(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.