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