Isomorphisms between Algebras of Semiclassical Pseudodifferential Operators

Following the work of Duistermaat-Singer [DS] on isomorphisms of algebras of global pseudodifferential operators, we classify isomorphisms of algebras of microlocally defined semiclassical pseudodifferential operators. Specifically, we show that any such isomorphism is given by conjugation by A = BF , where B is a microlocally elliptic semiclassical pseudodifferential operator, and F is a microlocal h-FIO associated to the graph of a local symplectic transformation.