On isomorphisms of algebras of smooth functions

Abstract:

We show that for any smooth Hausdorff manifolds $M$ and $N$, which are not necessarily second-countable, paracompact or connected, any isomorphism from the algebra of smooth (real or complex) functions on $N$ to the algebra of smooth functions on $M$ is given by composition with a unique diffeomorphism from $M$ to $N$. An analogous result holds true for isomorphisms of algebras of smooth functions with compact support.