Independence of the spectrum of an elliptic operator over a \(C^*\)- algebra (Q1071241)

Let \(A\) be a \(C^*\)-algebra and suppose \(\eta\) is an \(A\)-bundle over a closed compact smooth manifold. Consider an elliptic pseudo-differential operator \(\sigma(\partial/\partial x)\) acting on the Sobolev space \(H_ 0(\eta)\) [\textit{A. S. Mishchenko} and \textit{A. T. Fomenko}, Izv. Akad. Nauk. SSSR, Ser. Mat. 43, 831--859 (1979; Zbl 0416.46052)]. Given a representation \(\rho\) of \(A\), there is a natural Hilbert space operator \(\sigma_{\rho}\) associated with \(\sigma\). The author proves that the spectrum of \(\sigma_{\rho}\) is independent of the choice of \(\rho\). This generalizes earlier work in which \(\rho\) was assumed to be irreducible or the underlying Hilbert space for \(\rho (A)\) was assumed to be separable.