Sobolev algebras on Lie groups and noncommutative geometry

2022/03/13 by Arhancet, Cédric
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2203.06603

Abstract

We show that there exists a quantum compact metric space which underlies the setting of each Sobolev algebra associated to a subelliptic Laplacian Δ=-(X₁²+⋯+X_m²) on a compact connected Lie group G if p is large enough, more precisely under the (sharp) condition p > \fracdα where d is the local dimension of (G,X) and where 0 < α≤ 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.

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