Banach envelopes of $p$-Banach lattices, $0<p<1$, and Cesàro spaces

Abstract

In this note we characterize Banach envelopes of $p$-Banach lattices, $0<p<1$, such that their positive cones are $1$-concave. In particular we show that the Banach envelope of Cesàro sequence space $\widehat{ces_p(v)}$, $0<p<1$, coincides isometrically with the weighted $\ell_1(w)$ space where $w(n) = ||e_n||_{ces_p(v)}= (\sum_{i=n}^\infty i^{-p} v(i))^{1/p}$ and $e_n$ are the unit vectors. For Cesàro function space $Ces_p(v)$, $0<p<1$, its Banach envelope $\widehat{Ces_p(v)}$ is isometrically equal to $L_1(w)$ with $w(t) = (\int_t^\infty s^{-p} v(s) ds)^{1/p}$, $t\in (0,\infty)$.