On isomorphisms of algebras of smooth functions

TL;DR

This paper proves that for a wide class of smooth manifolds, any algebra isomorphism between their smooth functions corresponds uniquely to a diffeomorphism, extending previous results to non-second countable, non-paracompact, or disconnected manifolds.

Contribution

It establishes a general isomorphism-diffeomorphism correspondence for smooth function algebras on broad classes of manifolds, including non-second countable and disconnected cases.

Findings

Isomorphisms of smooth function algebras are induced by diffeomorphisms.

The result applies to manifolds without assumptions of second countability or connectedness.

Analogous results hold for algebras of smooth functions with compact support.

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.