Smoothness of inversion under subset diffeology for unit groups of diffeological algebras

Determine whether the inversion map a ↦ a^{-1}: A^* → A^* is diffeologically smooth when the unit group A^* of a diffeological algebra A is equipped with the subset diffeology inherited from A. Establish this smoothness property in full generality or identify precise conditions under which it holds, in order to clarify the relationship between the natural diffeology on A^* and the subset diffeology on A.

Background

In the paper, the author constructs diffeological tensor algebras of formal power series and proves smoothness of inversion in this specific setting using classical formal series expansions. However, the general situation for arbitrary diffeological algebras is more delicate: the natural diffeology on the unit group A* may differ from the subset diffeology inherited from the ambient algebra A.

The remark emphasizes that for certain diffeological algebras—such as the algebra L(V) of smooth linear endomorphisms of a diffeological vector space—the smoothness of inversion under the subset diffeology is not guaranteed in general. Resolving this uncertainty is significant for the theory of diffeological groups, since the smoothness of inversion is a key requirement for A* to be a diffeological group under the subset diffeology.

References

In general, when $A$ is a diffeological algebra, the natural diffeology of $A*$ can be different from the subset diffeology of $A,$ see e.g. for discussions on the diffeological algebra $L(V)$ of smooth linear endomorphisms of a diffeological vector space $V.$ Indeed, in general, we are not sure that the inversion map $a \mapsto a{-1}$ is smooth for the dubset diffeology.

— Diffeological Generalized Formal Series: An Overview  (2508.15786 - Magnot, 6 Aug 2025) in Remark after Theorem “tensor diffeological group”, Subsubsection “Diffeologies on power series of algebraic tensor product algebras” (Section 3.3.1)