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 holomorphic functions in non-commuting variables introduced by Dosi and apply our methods to prove theorems strengthening some his results. read more

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Functional Analysis