Local symmetry and smoothness in the space of vector-valued continuous functions

TL;DR

The paper investigates symmetry properties of vector-valued spaces with respect to Birkhoff-James orthogonality, extending Komuro's scalar results to $C(K,X)$ and establishing smoothness criteria in $C_0(K,X)$. It leverages norm-attainment sets $M_f=\{krom K:\|f(k) ight floor=\|f ight floor_\infty ight ight rait}$ and directional orthogonality, together with dual/extreme-point analyses, to derive necessary and sufficient conditions. Key contributions include a complete left symmetric classification in real $X$ (and in the complex case under sequential compactness), a norm-attainment-set based criterion for right symmetry, and a sharp smoothness criterion for $C_0(K,X)$, along with partial converses and counterexamples clarifying the scope. The work advances the geometric understanding of vector-valued function spaces and generalizes scalar results of Komuro and Sundaresan to broader settings with implications for Banach space geometry.

Abstract

In this article, we characterize the left symmetric points in $C(K,X)$, where $K$ is a compact Hausdorff space and $X$ is a Banach space. We also provide necessary and sufficient conditions for the right symmetric points in $C(K,X)$. Further, we identify the smooth points in the space $C_0(K,X)$, $K$ being locally compact Hausdorff space and $X$ being a Banach space.

Theorems & Definitions (18)

• Definition 1.1
• Theorem 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• Proposition 2.4
• proof
• Theorem 2.5
• proof
• Corollary 2.6