Isomorphisms of algebras of smooth functions revisited

TL;DR

This paper proves that algebra isomorphisms between smooth function algebras on Hausdorff manifolds are induced by diffeomorphisms, extending previous results without assumptions of second countability or paracompactness.

Contribution

It establishes that such algebra isomorphisms are always realized by diffeomorphisms, solving a problem posed by A. Wienstein and broadening the class of manifolds considered.

Findings

Isomorphisms correspond to diffeomorphisms.

No second countability or paracompactness needed.

Addresses a longstanding open problem.

Abstract

It is proved that isomorphisms between algebras of smooth functions on Hausdorff smooth manifolds are implemented by diffeomorphisms. It is not required that manifolds are second countable nor paracompact. This solves a problem stated by A. Wienstein. Some related results are discussed as well.