Functions of class $C^\infty$ in non-commuting variables in the context of triangular Lie algebras

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