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

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 characterizatio...

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Bibliographic Details
Published in:Journal of functional analysis Vol. 283; no. 1; p. 109502
Main Authors: Liu, Minzeng, Liu, Rui, Lu, Jimeng, Zheng, Bentuo
Format: Journal Article
Language:English
Published: Elsevier Inc 01.07.2022
Subjects:
Ball covering property
Complemented subspaces
Weak basis
π-basis
Ball covering property
Weak basis
π-basis
Complemented subspaces
ISSN:0022-1236, 1096-0783
Online Access:Get full text
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Summary: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 subspace containing finite rank operators in B(ℓp) for 1<p<∞ all have the BCP, and B(L1[0,1]) fails the BCP. Using those characterizations and results, we show that BCP is not hereditary for 1-complemented subspaces (even for completely 1-complemented subspace in operator space sense) by constructing two different counterexamples.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2022.109502