When a completion of the universal enveloping algebra is a Banach PI-algebra?

TL;DR

This paper characterizes when the completion of the universal enveloping algebra of a finite-dimensional Lie algebra is a Banach PI-algebra, linking polynomial identities to nilpotent radicals and growth conditions.

Contribution

It establishes a precise criterion involving nilpotent radicals and polynomial growth for the completion to satisfy a polynomial identity.

Findings

Banach algebra completion satisfies PI iff nilpotent radical is nilpotent in B

Polynomial growth condition on the radical characterizes PI property

Provides a link between algebraic structure and analytic properties

Abstract

We prove that a Banach algebra $B$ that is a completion of the universal enveloping algebra of a finite-dimensional complex Lie algebra $\mathfrak{g}$ satisfies a polynomial identity if and only if the nilpotent radical $\mathfrak{n}$ of $\mathfrak{g}$ is associatively nilpotent in $B$. Furthermore, this holds if and only if a certain polynomial growth condition is satisfied on $\mathfrak{n}$.