The \(C^*\)-algebra of the Mautner group

Let \(M_\theta =({\mathbb {R}} < imes {\mathbb {C}}^2, \underset{\theta }{\cdot }) \ (\theta \) $M_{\theta }=(\mathbb{R}⋉{\mathbb{C}}^2,\underset{\theta }{·})(\theta$ an irrational number), be the Mautner group. We describe the \(C^*\) $C^{\ast }$ -algebra of \(M_\theta \) $M_{\theta }$ as a subalgebra of \(C_0({\mathbb {C}}^2,{\mathcal {B}}(L^{2}({\mathbb {R}}))) \) $C_0({\mathbb{C}}^2,\mathcal{B}(L^2(\mathbb{R})))$