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

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 $\mathfrak{n}$.

version 3

References in corpus (3)

• Holomorphic functions of exponential type on connected complex Lie groups
• Length functions exponentially distorted on subgroups of complex Lie groups
• Decomposition of the algebra of analytic functionals on a connected complex Lie group and its completions into iterated analytic smash products

Cited by in corpus (1)

• Length functions exponentially distorted on subgroups of complex Lie groups