Sobolev algebras on Lie groups and noncommutative geometry

We show that there exists a quantum compact metric space which underlies the setting of each Sobolev algebra associated to a subelliptic Laplacian $\Delta=-(X\_1^2+\cdots+X\_m^2)$ on a compact connected Lie group $G$ if $p$ is large enough, more precisely under the (sharp) condition $p > \frac{d}{\alpha}$, where $d$ is the local dimension of $(G,X)$ and where $0 < \alpha \leq 1$. We also provide locally compact variants of this result and generalizations for real second-order subelliptic operators. We also introduce a compact spectral triple (= noncommutative manifold) canonically associated to each subelliptic Laplacian on a compact group. In addition, we show that its spectral dimension is equal to the local dimension of $(G,X)$. Finally, we prove that the Connes spectral pseudo-metric allows us to recover the Carnot–Carathéodory distance.