An algebraic approach to manifold-valued generalized functions

TL;DR

This paper explores the algebraic structure of manifold-valued generalized functions, establishing isomorphisms that connect algebraic and geometric perspectives, and extending classical results on smooth functions.

Contribution

It introduces an algebraic framework for manifold-valued generalized functions and proves isomorphisms that relate these to classical smooth function algebras.

Findings

Isomorphisms between Colombeau algebras and manifold-valued generalized functions

Extension of classical smooth function algebra results

Validation of the algebraic approach through consistency checks

Abstract

We discuss the nature of structure-preserving maps of varies function algebras. In particular, we identify isomorphisms between special Colombeau algebras on manifolds with invertible manifold-valued generalized functions in the case of smooth parametrization. As a consequence, and to underline the consistency and validity of this approach, we see that this generalized version on algebra isomorphisms in turn implies the classical result on algebras of smooth functions.