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), <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_421_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">$B(\ell ^p)/K(\ell ^p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>ℓ</mi> <mi>p</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>K</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>ℓ</mi> <mi>p</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_421_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1 \le p < \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>) and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_421_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">$B(c_0)/K(c_0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>K</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> all fail the BCP. We also show that <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_421_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$B(L^p[0,1])$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has the uniform ball-covering property (UBCP) for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_421_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$3/2 <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.

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

期刊论文
发表日期：

2025年4月19日

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Qiyao Bao,
Rui Liu,
Jie Shen

摘要

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