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

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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). 
@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},
     publisher = {mathdoc},
     volume = {73},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1977_73_a11/}
}
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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/