Copies of \(l^{\infty }\) in Quasi-normed Calderón–Lozanovskiĭ Spaces

We consider order isomorphic and order linearly isometric copies of \(l^{\infty }\) $l^{\infty }$ in the quasi-normed Calderón–Lozanovskiĭ spaces \(E_{\varphi }\) $E_{\phi }$ . We present a number of theorems describing these copies in the natural language of suitable properties of the quasi-normed ideal space E and the non-decreasing Orlicz function \(\varphi \) $\phi$ . In particular, we will characterize quasi-normed Orlicz–Lorentz spaces having order isomorphic and order linearly isometric copies of \(l^{\infty }\) $l^{\infty }$ . Our studies are conducted in a full possible generality due to the measure space and the Orlicz function.