Addendum to “uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces”

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 ℓ1(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{1}(X)$$\end{document} of a Banach space X has a unique unconditional basis whenever X does.

Dealing only with quasi-Banach spaces whose Banach envelope is isomorphic to 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 p ( q ), 0 < p, q ≤ ∞ (where ∞ means c 0 ) we realize that the p-Banach spaces p ( 2 ), p (c 0 ), and c 0 ( p ) for 0 < p < 1, have a unique unconditional basis (see [5,12]); since the Banach envelopes of those spaces, namely 1 ( 2 ), 1 (c 0 ) and c 0 ( 1 ) respectively, also do (see [6]), these examples reinforce the above-mentioned pattern.
Bourgain et al. proved that c 0 ( 2 ) has a unique unconditional basis but that, in contrast, the spaces 2 (c 0 ) and 2 ( 1 ) do not. We observe that while neither 2 (c 0 ) nor c 0 ( 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 2 ( p ) for 0 < p < 1 whose Banach envelope is 2 ( 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 T can be used because Casazza and Kalton [8] proved that that c 0 (T ) does not have a unique unconditional basis even though T does (see [7, Theorem 5.1]) . The original Tsirelson's space T * has also a unique unconditional basis. This can be deduced from the following result in combination with the fact that T is the dual space of T * (see [9]).

Lemma 1 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 (T ) gives that its dual space 1 (T * ) does not have a unique unconditional basis in spite of the fact that 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 1 -sums is raised. In addition, combining our remark with [1, Theorem 4.2] we solve in the negative the above-mentioned conjecture: Theorem 2 For each 0 < p < 1 there exists a p-Banach space X with a unique unconditional basis whose Banach envelope X does not have a unique unconditional basis.
For the sake of completeness we close this informative note by proving Lemma 1.

Proof of 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 assumption, they are permutatively equivalent. By the reflexivity principle for basic sequences in Banach spaces (see [3,Corollary 3.2.4]), X and Y are equivalent up to a permutation.