  For a locally compact group $G$, we show that it is possible to present the
class of continuous unitary representations of $G$ as an elementary class of
metric structures, in the sense of continuous logic. More precisely, we show
how non-degenerate $*$-representations of a general $*$-algebra $A$ (with some
mild assumptions) can be viewed as an elementary class, in a many-sorted
language, and use the correspondence between continuous unitary representations
of $G$ and non-degenerate $*$-representations of $L^1(G)$. We relate the notion
of ultraproduct of logical structures, under this presentation, with other
notions of ultraproduct of representations appearing in the literature, and
characterise property (T) for $G$ in terms of the definability of the sets of
fixed points of $L^1$ functions on $G$.
