Asymptotically isometric and isometric copies of ${\ell }_1$ in some Banach function lattices

We identify the class of Caldern-Lozanovskii spaces that do not contain an asymptotically isometric copy of ${\ell }_1$, and consequently we obtain the corresponding characterizations in the classes of Orlicz-Lorentz and Orlicz spaces equipped with the Luxemburg norm. We also give a~complete description of order continuous Orlicz-Lorentz spaces which contain (order) isometric copies of ${\ell }_1^{(n)}$ for each integer $n\ge 2$.~As an application we provide necessary and sufficient conditions for order continuous Orlicz-Lorentz spaces to contain an (order) isometric copy of ${\ell }_1$.~In particular we give criteria in Orlicz and Lorentz spaces for (order) isometric containment of ${\ell }_1^{(n)}$ and ${\ell }_1$.~The results are applied to obtain the~description of universal Orlicz-Lorentz spaces for all two-dimensional normed spaces.