Full and special Colombeau algebras

TL;DR

This paper introduces a unified framework for full and special Colombeau algebras with diffeomorphism invariance, defining point values, sheaf properties, and topology within this generalized setting.

Contribution

It unifies full and special Colombeau algebras into a single diffeomorphism-invariant framework with new definitions and properties.

Findings

Unified full and special Colombeau algebras with $ ext{ extbackslash epsilon}$-dependence

Characterization of elements using special generalized points

Conditions for sheaf property and sharp topology

Abstract

We introduce full diffeomorphism-invariant Colombeau algebras with added $\varepsilone$-dependence in the basic space. This unites the full and special settings of the theory into one single framework. Using locality conditions we find the appropriate definition of point values in full Colombeau algebras and show that special generalized points suffice to characterize elements of full Colombeau algebras. Moreover, we specify sufficient conditions for the sheaf property to hold and give a definition of the sharp topology in this framework.