Characterizations of universal finite representability and b-convexity of Banach spaces via ball coverings

Wen, Zhang

doi:10.1007/s10114-012-0721-z

Cited by 2 publications

(1 citation statement)

References 9 publications

(13 reference statements)

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“…Bentuo Zheng was supported in part by Simons Foundation Grant 585081. all ε > 0, which further reveals the fact that the BCP is a geometric property deeply related to the weak star topology for dual spaces. The BCP is also widely linked to a number of important properties, such as the G δ property of points in X, Radon-Nikodym property [4], uniform convexity, uniform non-squareness, strict convexity and dentability [13,14], and universal finite representability and B-convexity [16].…”

Section: Introductionmentioning

confidence: 99%

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

Minzeng¹,

Li²,

Jimeng³

et al. 2021

Preprint

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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 C 0 (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 C 0 (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(c 0 ), B(ℓ 1 ) and every subspaces containing finite rank operators in B(ℓ p ) for 1 < p < ∞ all have the BCP, and B(L 1 [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 subspaces in operator space sense) by constructing two different counterexamples.

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Section: Introductionmentioning

confidence: 99%