A Note on Symmetry of Birkhoff-James Orthogonality in Positive Cones of Locally C*-algebras

Abstract

In the present note some results of Kimuro, Saito, and Tanaka on symmetry of Birkhoff-James orthogonality in positive cones of C*-algebras are extended to locally C*-algebras.

Study on Orthogonal Sets for Birkhoff Orthogonality

“…Similarly, x ∈ X is called a right symmetric point if y ⊥ B x implies that x ⊥ B y for all y ∈ X. We note that in the study of symmetry in Birkhoff orthogonality, many scholars have focused on characterizing left symmetric points and right symmetric points in different types of normed spaces [7][8][9][10][11][12][13][14]. However, there exist x, y ∈ X, which are neither left symmetric points nor right symmetric points but satisfy x ⊥ B y and y ⊥ B x.…”