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

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 <= p Nicotiana benthamiana research toolbox through the generation of dicer-like mutants using CRISPR/Cas9 approaches