  For given triangular real Lie algebra $\mathfrak{g}$ we construct a certain
completion $C^\infty(\mathfrak{g})$ of its universal enveloping algebra. It is
a Fr\'echet-Arens-Michael algebra, which consists of elements of polynomial
growth and satisfies to the following universal property: every Lie algebra
homomorphism from $\mathfrak{g}$ to a real Banach algebra all of whose elements
are of polynomial growth has an extension to a continuous homomorphism from
$C^\infty(\mathfrak{g})$. Elements of this algebra can be called functions of
class $C^\infty$ in non-commuting variables. The proof is based on
representation theory and uses an ordered $C^\infty$-functional calculus.
Beyond the general case, we analyze two simple examples. As an auxiliary
material, we develop the basics of the general theory of algebras of polynomial
growth. We also consider local variants of the completion and obtain conditions
for existence of a sheaf of non-commutative functions on the Gelfand spectrum
of $C^\infty(\mathfrak{g})$. In addition, we discuss the theory of holonorphic
functions in non-commuting variables introduced by Dosi and apply our methods
to prove theorems strengthening some his results.
