Differential operators over \(C^*\)-algebras (Q1293292)

Let \(M\) be a closed smooth manifold, \(A\) be a unital \(C^*\)-algebra. The Sobolev space \(W_k(V)\) for an \(A\)-module bundle \(V\) over a manifold with boundary \(M\) consists of \({\mathcal L}^2\) sections of \(V\) with \(k\) distributional derivatives in \({\mathcal L}^2\) \((k>0)\). Let \(W^0_k(V)\) denote the subspace satisfying zero Dirichlet boundary conditions, \(D_0\) be a formally self-adjoint \(A\)-linear elliptic differential operator and \(D\) be its extension with domain \({\mathcal D}(D)= F_{2k}(V):= W_{2k}(V)\cap W^0_k(V)\). It is shown that the Sobolev inner products \((\cdot,\cdot)_k\) give rise to \(D\) on \(F_{2k}(V)\) with strictly positive spectrum. The domain definition of \(D^{1/2}\) is \(W^0_k(V)\) and there \((\cdot,\cdot)_k= (\cdot,\cdot)_+\) which is defined as \((D^{1/2}\cdot, D^{1/2}\cdot)\). It follows that the Sobolev inner products are all compatible. This gives a preferred compatibility class of inner products on \(W^0_k(V)\), depending only on the smooth structure of \(M\). As consequences, differential or properly supported pseudodifferential operators between Sobolev space have adjoints, and Rellich's theorem holds.