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

TL;DR

This paper constructs a universal algebra of non-commutative $C^infty$ functions associated with triangular Lie algebras, employing representation theory and functional calculus, and explores its properties and examples.

Contribution

It introduces a new completion of the universal enveloping algebra as a $C^infty$-function algebra in non-commuting variables, extending the theory with examples and local variants.

Findings

Constructed a Fre9chet-Arens-Michael algebra with a universal property.

Developed a sheaf of non-commutative functions on the Gelfand spectrum for nilpotent cases.

Strengthened existing theorems on holomorphic functions in non-commuting variables.

Abstract

We construct a certain completion $C^\infty_\mathfrak{g}$ of the universal enveloping algebra of a triangular real Lie algebra $\mathfrak{g}$. It is a Fr\'echet-Arens-Michael algebra that 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 with domain~$C^\infty_\mathfrak{g}$. Elements of this algebra can be called functions of class $C^\infty$ in non-commuting variables. The proof is based on representation theory and employs an ordered $C^\infty$-functional calculus. Beyond the general case, we analyze two simple examples. As an auxiliary material, the basics of the general theory of algebras of polynomial growth are developed. We also consider local variants of the completion and obtain a sheaf of non-commutative functions on the Gelfand spectrum of $C^\infty_\mathfrak{g}$ in the case when $\mathfrak{g}$ is nilpotent. 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.