Birkhoff-James Orthogonality and Its Pointwise Symmetry in Some Function Spaces

Abstract

We study Birkhoff-James orthogonality and its pointwise symmetry in commutative $C^{\ast }$ algebras, i.e., the space of all continuous functions defined on a locally compact Hausdorff space that vanish at infinity. We use this characterization to obtain the characterization of Birkhoff-James orthogonality on $L_{\infty }$ space defined on any arbitrary measure space. We also do the same for the $L_p$ spaces for $1\le p<\infty$.