Abstract
After [Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces, Positivity 26 (2022), Paper no. 35] was published, we realized that Theorem 4.2 therein, when combined with work of Casazza and Kalton (Israel J. Math. 103:141–175, 1998) , solves the long-standing problem whether there exists a quasi-Banach space with a unique unconditional basis whose Banach envelope does not have a unique unconditional basis. Here we give examples to prove that the answer is positive. We also use auxiliary results in the aforementioned paper to give a negative answer to the question of Bourgain et al. (Mem Am Math Soc 54:iv+111, 1985)*Problem 1.11 whether the infinite direct sum \(\ell _{1}(X)\) of a Banach space X has a unique unconditional basis whenever X does.
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Roughly speaking, the Banach envelope \({\widehat{X}}\) of a quasi-Banach space X is the Banach space “closest” to X. It is not surprising then that \({\widehat{X}}\) and X share many important structural features such as having the same dual space. If \({\mathcal {X}}\) is a normalized unconditional basis of X then \({\mathcal {X}}\) is a semi-normalized unconditional basis of \({\widehat{X}}\) (see [2, Section 10]) so it is natural to wonder if the property of having a unique unconditional basis (up to equivalence and permutation) will be transferred to the Banach envelope. This problem is far from trivial since in all known spaces so far, the pattern shows that \({\widehat{X}}\) has a unique unconditional basis whenever X does. Take, for instance, the classical p-Banach spaces \(\ell _p\), \(H_p({\mathbb {T}})\), \(\ell _p(\ell _1)\) or \(\ell _1(\ell _p)\) for \(0<p<1\), which have a unique unconditional basis (see [4, 10, 14]) and whose Banach envelope \(\ell _1\) also does [13].
Dealing only with quasi-Banach spaces whose Banach envelope is isomorphic to \(\ell _1\) is too restrictive. To obtain a better insight into the underlying pattern, we must look at quasi-Banach spaces with a more complicated Banach envelope. If we focus on the mixed-norm matrix spaces \(\ell _p(\ell _q)\), \(0<p,q\le \infty \) (where \(\ell _{\infty }\) means \(c_{0}\)) we realize that the p-Banach spaces \(\ell _p(\ell _2)\), \(\ell _p(c_0)\), and \(c_0(\ell _p)\) for \(0<p<1\), have a unique unconditional basis (see [5, 12]); since the Banach envelopes of those spaces, namely \(\ell _1(\ell _2)\), \(\ell _1(c_0)\) and \(c_0(\ell _1)\) respectively, also do (see [6]), these examples reinforce the above-mentioned pattern.
Bourgain et al. proved that \(c_0(\ell _2)\) has a unique unconditional basis but that, in contrast, the spaces \(\ell _2(c_0)\) and \(\ell _2(\ell _1)\) do not. We observe that while neither \(\ell _2(c_0)\) nor \(c_0(\ell _2)\) are the Banach envelope of a non-locally convex natural quasi-Banach space with a basis [11], there are non-locally convex spaces such as \(\ell _2(\ell _p)\) for \(0<p<1\) whose Banach envelope is \(\ell _2(\ell _1)\). However, no technique specific to non-locally convex spaces has been shown to be effective to determine whether these spaces have a unique unconditional basis.
Classical Banach spaces seem not to provide examples that disprove the conjecture that uniqueness of unconditional basis passes to Banach envelopes, but the non-classical Tsirelson space \({\mathcal {T}}\) can be used because Casazza and Kalton [8] proved that that \(c_0({\mathcal {T}})\) does not have a unique unconditional basis even though \({\mathcal {T}}\) does (see [7, Theorem 5.1]) . The original Tsirelson’s space \({\mathcal {T}}^*\) has also a unique unconditional basis. This can be deduced from the following result in combination with the fact that \({\mathcal {T}}\) is the dual space of \({\mathcal {T}}^*\) (see [9]).
Let X be a Banach space with an unconditional basis. Suppose that \(X^*\) has a unique unconditional basis. Then X has a unique unconditional basis too.
Applying Lemma 1 with \(X=c_0({\mathcal {T}})\) gives that its dual space \(\ell _1({\mathcal {T}}^*)\) does not have a unique unconditional basis in spite of the fact that \({\mathcal {T}}^*\) does. Notice the tight connection of this example to [6, Problem 11.1], where the question of whether the uniqueness of unconditional basis passes to infinite \(\ell _{1}\)-sums is raised. In addition, combining our remark with [1, Theorem 4.2] we solve in the negative the above-mentioned conjecture:
For each \(0<p<1\) there exists a p-Banach space X with a unique unconditional basis whose Banach envelope \({\widehat{X}}\) does not have a unique unconditional basis.
Indeed, the p-Banach space \(\ell _p({\mathcal {T}}^*)\), \(0<p<1\), has a unique unconditional basis (see [1, Example 7.12(ii)]), and its Banach envelope is \(\ell _1({\mathcal {T}}^*)\).
For the sake of completeness we close this informative note by proving Lemma 1.
Since \(X^*\) has a basis, it is separable. Therefore, since the property of having a separable dual is inherited by subspaces, X contains no isomorphic copy of \(\ell _1\). Then, by [3, Corollary 3.3.3], any unconditional basis of X is shrinking. Let \({\mathcal {X}}\) and \({\mathcal {Y}}\) be normalized unconditional bases of X. The basic sequences \({\mathcal {X}}^*\) and \({\mathcal {Y}}^*\) of their biorthogonal functionals are semi-normalized unconditional bases of \(X^*\). Hence, by assumption, they are permutatively equivalent. By the reflexivity principle for basic sequences in Banach spaces (see [3, Corollary 3.2.4]), \({\mathcal {X}}\) and \({\mathcal {Y}}\) are equivalent up to a permutation.
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Acknowledgements
The authors would like to thank the editor for his encouragement to publish this addendum and for his helpful comments.
Funding
Open Access funding provided by Universidad Pública de Navarra. F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación
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Albiac, F., Ansorena, J.L. Addendum to “uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces”. Positivity 26, 78 (2022). https://doi.org/10.1007/s11117-022-00942-w
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DOI: https://doi.org/10.1007/s11117-022-00942-w