Envelopes in the class of Banach algebras of polynomial growth and $C^{\mathrm{\infty }}$-functions of a finite number of free variables

We introduce the notion of envelope of a topological algebra (in particular, an arbitrary associative algebra) with respect to a class of Banach algebras. In the case of the class of real Banach algebras of polynomial growth, i.e., admitting a $C^{\mathrm{\infty }}$-functional calculus for every element, we get a functor that maps the algebra of polynomials in $k$ variables to the algebra of $C^{\mathrm{\infty }}$-functions on ${\mathbb{R}}^k$. The envelope of a general commutative or non-commutative algebra can be treated as an algebra of $C^{\mathrm{\infty }}$-functions on some commutative or non-commutative space. In particular, we describe the envelopes of the universal enveloping algebra of finite-dimensional Lie algebras, the coordinate algebras of the quantum plane and quantum $SL(2)$ and also look at some commutative examples. A result on algebras of `free $C^{\mathrm{\infty }}$-functions’, i.e., the envelopes of free associative algebras of finite rank $k$, is announced for general $k$ and proved for $k\le 2$.