Ball covering property from commutative function spaces to non-commutative spaces of operators

2021/11/09 by Minzeng Liu, Rui Liu, Liu, Minzeng +5 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2111.04921

openalex publication_date 2021/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A Banach space is said to have the ball-covering property (abbreviated BCP) if its unit sphere can be covered by countably many closed, or equivalently, open balls off the origin. Let K be a locally compact Hausdorff space and X be a Banach space. In this paper, we give a topological characterization of BCP, that is, the continuous function space C₀(K) has the (uniform) BCP if and only if K has a countable π-basis. Moreover, we give the stability theorem: the vector-valued continuous function space C₀(K,X) has the (strong or uniform) BCP if and only if K has a countable π-basis and X has the (strong or uniform) BCP. We also explore more examples for BCP on non-commutative spaces of operators B(X,Y). In particular, these results imply that B(c₀), B(ℓ₁) and every subspaces containing finite rank operators in B(ℓ_p) for 1< p

Cited by

• Approximation properties and quantitative estimation for uniform ball-covering property of operator spaces

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