  We introduce the notion of envelope of a topological algebra (in particular,
an arbitrary associative algebra) with respect to a class of Banach algebra. In
the case of the class of real Banach algebras of polynomial growth, i.e.,
admitting a $C^\infty$-functional calculus for every element, we get a functor
that maps the algebra of polynomials in $k$ variables to the algebra of
$C^\infty$-functions on $\mathbb{R}^k$. The envelope of a general commutative
or non-commutative algebra can be treated as an algebra of $C^\infty$-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
$C^\infty$-functions', i.e., the envelopes of free associative algebras of
finite rank $k$, is announced for general $k$ and proved for $k\le 2$.
