Geodesic
Space of operators acting from one banach lattice to another
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 188-192 
Yu. A. Abramovich. Space of operators acting from one banach lattice to another. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 188-192. http://geodesic.mathdoc.fr/item/ZNSL_1977_73_a11/
@article{ZNSL_1977_73_a11,
     author = {Yu. A. Abramovich},
     title = {Space of operators acting from one banach lattice to another},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {188--192},
     year = {1977},
     volume = {73},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1977_73_a11/}
}
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In this note we construct a pair of Banach lattices $X$ and $Y$, which have the following properties: a) $X$ is not order isomorphic to an $AL$-space, b) $Y$ is not order isomorphic to an $AM$-space, c) for any continuous linear operator $T:X\to Y$ there exists a modulus $|T|:X\to Y$. This example refutes the conjecture of Cartwright–Lotz, saying that the negation of at least one of the conditions a) or b) is necessary for the validity of c).