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

TL;DR

This paper characterizes Birkhoff-James orthogonality and its pointwise symmetry in function spaces, including continuous functions and various $L_p$ spaces, providing new insights into their geometric properties.

Contribution

It offers novel characterizations of Birkhoff-James orthogonality in commutative $C^*$ algebras and $L_p$ spaces, extending understanding of orthogonality in these function spaces.

Findings

Characterization of Birkhoff-James orthogonality in $C^*$ algebras

Extension of orthogonality characterization to $L_ ext{infty}$ spaces

Extension of orthogonality characterization to $L_p$ spaces for $1 \,\leq p<\infty$

Abstract

We study Birkhoff-James orthogonality and its pointwise symmetry in commutative $C^*$ 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 $1\leq p<\infty$.