On an isomorphism of the algebra of pseudo-differential operators (Q801582)

On the basis of a previous result by \textit{J. J. Duistermaat} and \textit{I. M. Singer} [Commun. Pure Appl. Math. 29, 39-47 (1976; Zbl 0317.58017)], the following theorem is proved. Let M be a compact connected smooth manifold, and \(\alpha\) an automorphism of the algebra of all pseudodifferential operators on M; assuming that (i) the two-sided ideal of operators with smooth kernel is globally invariant under \(\alpha\), and (ii) \(\alpha\) maps vector fields into vector fields, then \(\alpha\) is the isomorphism naturally defined by a (unique) diffeomorphism of M onto itself.