On Multiplicative Maps of Continuous and Smooth Functions

TL;DR

This paper characterizes multiplicative bijections on various function spaces over manifolds, showing they are mostly induced by diffeomorphisms and, in the complex case, include complex conjugation, with applications to Fourier transforms.

Contribution

It extends known theorems to describe the structure of multiplicative maps on continuous and smooth functions, including complex-valued cases and Fourier transform applications.

Findings

In the real case, multiplicative bijections are compositions with diffeomorphisms.

In the complex case, complex conjugation is the only additional symmetry.

Results apply to characterizing Fourier transforms between function spaces.

Abstract

In this note, we study the general form of a multiplicative bijection on several families of functions defined on manifolds, both real or complex valued. In the real case, we prove that it is essentially defined by a composition with a diffeomorphism of the underlying manifold (with a bit more freedom in families of continuous functions). Our results in the real case are mostly simple extensions of known theorems. We then show that in the complex case, the only additional freedom allowed is complex conjugation. Finally, we apply those results to characterize the Fourier transform between certain function spaces.