The ball-covering property of non-commutative spaces of operators on Banach spaces

2024/12/10 by Rui Liu, Bao, Qiyao, Jie Shen +2 · 1 citation
Mathematics · #Advanced Banach Space Theory #advanced mathematical theories #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2412.07137

Abstract

A Banach space is said to have the ball-covering property (BCP) if its unit sphere can be covered by countably many closed or open balls off the origin. Let X be a Banach space with a shrinking 1-unconditional basis. In this paper, by constructing an equivalent norm on B(X), we prove that the quotient Banach algebra B(X)/K(X) fails the BCP. In particular, the result implies that the Calkin algebra B(H)/ K(H), B(ℓ^p)/K(ℓ^p) (1 ≤ p <∞) and B(c₀)/K(c₀) all fail the BCP. We also show that B(L^p[0,1]) has the uniform ball-covering property (UBCP) for 3/2< p < 3.

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

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