Albiac Alesanco, Fernando JoséAnsorena, José L.2023-04-172023-04-172022Albiac, 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-w1385-129210.1007/s11117-022-00942-whttps://academica-e.unavarra.es/handle/2454/45133Adenda al artículo en https://hdl.handle.net/2454/43360After [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 l(1)(X) of a Banach space X has a unique unconditional basis whenever X does.application/pdfeng© The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License.Uniqueness of unconditional basisQuasi-Banach spaceArtículo / Artikulua2023-04-17Acceso abierto / Sarbide irekiainfo:eu-repo/semantics/openAccess