On unitary representations of algebraic groups over local fields

Let $\mathbf{G}$ be an algebraic group over a local field $\mathbf k$ of characteristic zero. We show that the locally compact group $\mathbf G(\mathbf k)$ consisting of the $\mathbf k$-rational points of $\mathbf G$ is of type I. Moreover, we complete Lipsman's characterization of the groups $\mathbf G$ for which every irreducible unitary representation of $\mathbf G(\mathbf k)$ is a CCR representation and show at the same time that such groups $\mathbf G(\mathbf k)$ are trace class as studied recently by Deitmar and van Dijk.