Uniqueness of unconditional basis of \(\ell_{p}(c_{0})\) and \(\ell_{p}(\ell_{2})\), \(0<p<1\) (Q2773428)

Let \(X\) be a quasi-Banach space with a normalized unconditional basis \((e_n)\). If for any normalized unconditional basis \((x_n)\) of \(X\) there exists a permutation \(\pi\) of \(\mathbb{N}\) such that \((x_n)\) is equivalent to \((e_{\pi(n)})\), then \(X\) is said to have a unique unconditional basis up to permutation. The Banach envelope \(\widehat X\) of \(X\) is defined as the completion of \((X,\|\cdot\|_C)\), where \(\|\cdot \|_C\) is the Minkowski functional of \(C\), the convex hull of the closed unit ball in \(X\). For example, \(\ell_1(c_0)\) and \(\ell_1(\ell_2)\) are the Banach envelopes of the quasi-Banach spaces \(\ell_p(c_0)\) and \(\ell_p(\ell_2)\), respectively, for \(0<p< 1\).NEWLINENEWLINENEWLINEIn 1985 \textit{J. Bourgain}, \textit{P. G. Casazza}, \textit{J. Lindenstrauss} and \textit{L. Tzafriri} [Mem. Am. Math. Soc. 322 (1985; Zbl 0575.46011)] have shown that in the Banach spaces \(\ell_1(c_0)\) and \(\ell_1(\ell_2)\) the canonical unit vector basis is unique up to permutation. In the present paper, the authors prove that the non-locally convex quasi-Banach spaces \(\ell_p(c_0)\) and \(\ell_p (\ell_2)\) \((0<p<1)\) also have a unique unconditional basis up to permutation.