Unitary Representations of the Isometry Groups of Urysohn Spaces

We obtain a complete classification of the continuous unitary representations of the isometry group of the rational Urysohn space $\mathbb{Q}\mathbb{U}$. As a consequence, we show that Isom$(\mathbb{Q}\mathbb{U})$ has property (T). We also derive several ergodic theoretic consequences from this classification: $(i)$ every probability measure-preserving action of Isom$(\mathbb{Q}\mathbb{U})$ is either essentially free or essentially transitive, $(ii)$ every ergodic Isom$(\mathbb{Q}\mathbb{U})$-invariant probability measure on $[0,1]^{\mathbb{Q}\mathbb{U}}$ is a product measure. We obtain the same results for isometry groups of variations of $\mathbb{Q}\mathbb{U}$, such as the rational Urysohn sphere $\mathbb{Q}\mathbb{U}_1$, the integral Urysohn space $\mathbb{Z}\mathbb{U}$, etc.