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

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 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.