Algebras of free $C^\infty$-functions

The algebra of $C^\infty$ real functions on a manifold is a quotient of $C^\infty(\mathbb{R}^k)$ for some $k$. One of the options for including spaces with singularities in the game is to consider arbitrary quotients ($C^\infty$-differentiable algebras). It follows from classical results that $C^\infty(\mathbb{R}^k)$ is a free algebra in the category of commutative real Arens-Michael algebras that have polynomial growth locally (PGL algebras). Our main goal is to describe non-commutative analogues — algebras of free $C^\infty$-functions, which are defined as PGL envelopes of free algebras of non-commutative polynomials. In the case of finitely many generators, this description provides us some information on quotients of algebras free $C^\infty$-functions, which we call `finitely $C^\infty$-generated algebras’. This notion is not only a non-commutative version of the concept of $C^\infty$-differentiable algebra but also a $C^\infty$ analogue of the concept of holomorphically finitely generated algebra introduced by Pirkovskii. One of the reasons for the interest in finitely $C^\infty$-generated algebras is their good behaviour with respect to the projective tensor product, which makes it possible to study topological Hopf algebras with underlying algebras in this class.