Metric groups, unitary representations and continuous logic

summary: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