Sobolev algebras on Lie groups and noncommutative geometry (Q6123804)

Summary: We show that there exists a quantum compact metric space which underlies the setting of each Sobolev algebra associated to a subelliptic Laplacian \(\Delta =-(X^2_1 +\cdots +X^2_m)\) 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 \(0<\alpha\leqslant 1\). 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.