A generalization of Drewnowski's result on the Cantor-Bernstein type theorem for a~class of nonseparable Banach spaces

Abstract

Let $X_a$ denote the order continuous part of a Banach lattice $X$, and let $\Gamma$ be an uncountable set. We extend Drewnowski's theorem on the comparison of linear dimensions between Banach spaces having uncountable symmetric bases to a~class of discrete Banach lattices, the so-called $D$-spaces. We show that if $X$ and $Y$ are two $D$-spaces and there are continuous linear injections (not necessarily embeddings) from $X$ into $Y$ and vice versa, then $X$ and $Y$ are order-topologically isomorphic. In the proof we apply a theorem on the extension of an order isomorphism from $X_a$ onto $Y_a$ to an order isomorphism from $X$ onto $Y$, the classical Drewnowski's theorem, and a supplement of Troyanski's theorem on embeddings of $\ell_1(\Gamma)$ spaces into a Banach space with an uncountable symmetric basis. Our result applies to the class of Orlicz spaces $\ell_\varphi(\Gamma)$, where $\varphi$ is an Orlicz function.