Geodesic
Metric groups, unitary representations and continuous logic
Communications in Mathematics, Tome 29 (2021) no. 1, pp. 35-48.

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We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find $L_{\omega _1 \omega }$-axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan's property (T) can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.
Classification : 03C52, 22F05
Keywords: Continuous logic; metric groups; unitary representations; amenable groups. 
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Ivanov, Aleksander. Metric groups, unitary representations and continuous logic. Communications in Mathematics, Tome 29 (2021) no. 1, pp. 35-48. http://geodesic.mathdoc.fr/item/COMIM_2021__29_1_a2/