Partial Algebras of Sobolev Spaces and their Groupoid Characterization

The partial multiplication by weak product and the weak derivative operation delineate the Sobolev spaces Wk,p(Ω) as locally convex partial ^*-algebras on Lᵖ(Ω). We characterised them as locally convex Lie groupoids W := Wk,p(Ω) Lᵖ(Ω). As invariant spaces of the convolution actions of the smooth algebra K(Ω) on the Banach spaces Lᵖ(Ω), the resulting Lie groupoids W is shown to have a unitary representation on a Hilbert bundle and a ^*-representation on its C^*-algebra.