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

TL;DR

This paper investigates the ball-covering property in non-commutative operator spaces on Banach spaces, showing that certain quotient algebras fail this property while others have the uniform ball-covering property.

Contribution

It constructs an equivalent norm on operator algebras to demonstrate failure of the BCP in specific quotient spaces, extending known results to new classes of Banach spaces.

Findings

The quotient algebra B(X)/K(X) fails the BCP for Banach spaces with a shrinking 1-unconditional basis.

The Calkin algebra B(H)/K(H) and similar spaces also fail the BCP.

B(L^p[0,1]) has the UBCP for 3/2< p < 3.

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 $B(H)/ K(H)$, $B(\ell^p)/K(\ell^p)$ ($1 \leq p <\infty$) and $B(c_0)/K(c_0)$ all fail the BCP. We also show that $B(L^p[0,1])$ has the uniform ball-covering property (UBCP) for $3/2< p < 3$.