Ball-covering property of Banach spaces that is not preserved under linear isomorphisms.

*(English)*Zbl 1152.46010
Sci. China, Ser. A 51, No. 1, 143-147 (2008); erratum Isr. J. Math. 184, 505-507 (2011).

The authors continue research that originated in the first author’s paper [L.-X. Cheng, Isr. J. Math. 156, 111–123 (2006; Zbl 1139.46016)]. A Banach space \(X\) has the ball-covering property if the unit sphere of \(X\) can be covered by a countable collection of balls not containing the origin. It was shown in the above cited paper that \(\ell_\infty\) in its standard norm \(\| \cdot\| _\infty\) has this property.

In the paper under review, the authors demonstrate by means of a short and elegant proof that \(\ell_\infty\) does not possess the ball-covering property in the equivalent norm \(\| \cdot\| := \| \cdot\| _\infty + p(\cdot)\), where \(p(x):= \lim\sup_n| x(n)| \). This means that the ball-covering property is not stable under isomorphisms. It is remarked also that \(\ell_\infty/c_0\) does not have the ball-covering property, which means that the ball-covering property is not inherited by quotients. Finally, the authors state that the ball-covering property is not inherited by subspaces. The proof is based on the existence of an isometric embedding of \((\ell_\infty, \| \cdot\| )\) into \((\ell_\infty, \| \cdot\| _\infty)\). Unfortunately, the last statement is false, which leaves the question open whether the ball-covering property is inherited by subspaces. The mistake is that the authors think erroneously that weak-star separability of \(X^*\) implies weak-star separability of \(B_{X^*}\).

Let us explain why \(X=(\ell_\infty, \| \cdot\| )\) cannot be isometrically embedded into \((\ell_\infty, \| \cdot\| _\infty)\) (or equivalently, there is no 1-norming sequence in \(X^*\)). Assume to the contrary that there is a sequence \(e_n^* \in B_{X^*}\) such that \(\sup_n| e_n^*(x)| = \| x\| \) for every \(x \in X\). Let us write \(e_n^*\) as \(f_n + g_n\), where \(f_n \in \ell_1 \subset \ell_\infty^*\) and \(g_n \in (c_0)^{\bot} \subset \ell_\infty^*\). Noting that on \(c_0\) the norm of \(X\) coincides with the standard norm \(\| \cdot\| _\infty\), consequently,

\[ 1 \geq \| e_n^*\| \geq \sup_{x \in B_{c_0}}| e_n^*(x)| =\sup_{x \in B_{c_0}}| f_n(x)| = \| f_n\| _{\ell_1}. \]

On the other hand, the standard fact that \(\ell_\infty/c_0\) does not admit a countable total system of functionals implies that there is an element \(z \in X \setminus c_0\) such that \(g_n(z)=0\) for all \(n\). For this \(z\) we have

\[ \| z\| = \sup_n| e_n^*(z)| = \sup_n| f_n(z)| \leq \| z\| _\infty, \]

i.e., \(p(z)=0\) and \(z \in c_0\), a contradiction.

In the paper under review, the authors demonstrate by means of a short and elegant proof that \(\ell_\infty\) does not possess the ball-covering property in the equivalent norm \(\| \cdot\| := \| \cdot\| _\infty + p(\cdot)\), where \(p(x):= \lim\sup_n| x(n)| \). This means that the ball-covering property is not stable under isomorphisms. It is remarked also that \(\ell_\infty/c_0\) does not have the ball-covering property, which means that the ball-covering property is not inherited by quotients. Finally, the authors state that the ball-covering property is not inherited by subspaces. The proof is based on the existence of an isometric embedding of \((\ell_\infty, \| \cdot\| )\) into \((\ell_\infty, \| \cdot\| _\infty)\). Unfortunately, the last statement is false, which leaves the question open whether the ball-covering property is inherited by subspaces. The mistake is that the authors think erroneously that weak-star separability of \(X^*\) implies weak-star separability of \(B_{X^*}\).

Let us explain why \(X=(\ell_\infty, \| \cdot\| )\) cannot be isometrically embedded into \((\ell_\infty, \| \cdot\| _\infty)\) (or equivalently, there is no 1-norming sequence in \(X^*\)). Assume to the contrary that there is a sequence \(e_n^* \in B_{X^*}\) such that \(\sup_n| e_n^*(x)| = \| x\| \) for every \(x \in X\). Let us write \(e_n^*\) as \(f_n + g_n\), where \(f_n \in \ell_1 \subset \ell_\infty^*\) and \(g_n \in (c_0)^{\bot} \subset \ell_\infty^*\). Noting that on \(c_0\) the norm of \(X\) coincides with the standard norm \(\| \cdot\| _\infty\), consequently,

\[ 1 \geq \| e_n^*\| \geq \sup_{x \in B_{c_0}}| e_n^*(x)| =\sup_{x \in B_{c_0}}| f_n(x)| = \| f_n\| _{\ell_1}. \]

On the other hand, the standard fact that \(\ell_\infty/c_0\) does not admit a countable total system of functionals implies that there is an element \(z \in X \setminus c_0\) such that \(g_n(z)=0\) for all \(n\). For this \(z\) we have

\[ \| z\| = \sup_n| e_n^*(z)| = \sup_n| f_n(z)| \leq \| z\| _\infty, \]

i.e., \(p(z)=0\) and \(z \in c_0\), a contradiction.

Reviewer: Vladimir Kadets (Kharkov)

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\textit{L. Cheng} et al., Sci. China, Ser. A 51, No. 1, 143--147 (2008; Zbl 1152.46010)

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