Norm closure and extension of the symbolic calculus for the cone algebra (Q1907146)

The paper deals with the `cone algebra' \({\mathfrak C}= {\mathfrak C}^0(\mathbb{B}, {\mathfrak g}^\bullet_\gamma)\) of B.-W. Schulze. This is a *-algebra of zero-order pseudodifferential operators on a (stretched) manifold \(\mathbb{B}\) with conical singularities; each \(A\in {\mathfrak C}\) acts on a scale \(\{{\mathcal H}^{s, \gamma}(\mathbb{B})\}_{s\in \mathbb{R}}\) of weighted \(L_2\)-Sobolev spaces. The symbol of \(A\) contains a scalar part \(\sigma_\psi(A)\) and an operator function \(\sigma_M(A)\), the `Mellin symbol'. The author studies the closures \({\mathfrak A}_s\) and \({\mathfrak A}_\infty\) of \(\mathfrak C\) in \({\mathcal L}({\mathcal H}^{s, \gamma}(\mathbb{B}))\) and \(\bigcap_s {\mathcal L}({\mathcal H}^{s, \gamma}(\mathbb{B}))\), respectively. These sets turn out to be \(C^*\)-resp. Fréchet\(^*\)-algebras, equipped with natural extensions of \(_s\sigma_{{\mathfrak C}}\), \(_\infty\sigma_{{\mathfrak C}}\) of the symbol homomorphism \(\sigma_{{\mathfrak C}}= (\sigma_\psi, \sigma_M)\). The Fredholm property of \(A\in {\mathfrak A}_\times\) is characterized by the invertibility of \(_\times\sigma_{{\mathfrak C}}(A)\). A crucial step is to show that the symbol algebras \(_s\sigma_{{\mathfrak C}}({\mathfrak A}_s)\), \(_\infty\sigma_{{\mathfrak C}}({\mathcal A}_\infty)\) are again of type \(C^*\) resp. Fréchet\(^*\). As a central result one obtains spectral invariance of \({\mathfrak A}_\infty\) in \({\mathcal L}({\mathcal H}^{s, \gamma}(\mathbb{B}))\); one of the consequences is the existence of a holomorphic functional calculus for \({\mathfrak A}_\infty\). The methods employed are easily adaptable to other pseudodifferential algebras; previous approaches of H. O. Cordes and E. Schrohe for algebras with scalar symbol seem to be less flexible. In a forthcoming paper an axiomatic outline of the present approach will be given.