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

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 ₁ (X) ℓ 1 (X) of a Banach space X has a unique unconditional basis whenever X does.