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We prove that a Banach algebra $B$ that is a completion of the universalenveloping 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 issatisfied on $\mathfrak{n}$.

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