Geodesic
A Hilbert $C^*$-modules with extremal properties
Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 1, pp. 94-105

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We construct an example of a Hilbert $C^*$-module which shows that Troitsky's theorem on the geometric essence of $\mathcal{A}$-compact operators between Hilbert $C^*$-modules cannot be extended to modules which are not countably generated case (even in the case of a stronger uniform structure, which is also introduced). In addition, the constructed module admits no frames.
Keywords: Hilbert $C^*$-module, uniform structure, totally bounded set, compact operator, $\mathcal{A}$-compact operator, locally compact space, Radon measure. 
@article{FAA_2022_56_1_a6,
     author = {D. V. Fufaev},
     title = {A {Hilbert} $C^*$-modules with extremal properties},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {94--105},
     publisher = {mathdoc},
     volume = {56},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2022_56_1_a6/}
}
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D. V. Fufaev. A Hilbert $C^*$-modules with extremal properties. Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 1, pp. 94-105. http://geodesic.mathdoc.fr/item/FAA_2022_56_1_a6/