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 $\mathfrak {n}$ .

Given a property of representations satisfying a basic stability condition, Ramakrishna developed a variant of Mazur’s Galois deformation theory for representations with that property. We introduce an axiomatic definition of pseudorepresentations with such a property. Among other things, we show that pseudorepresentations with a property enjoy a good deformation theory, generalizing Ramakrishna’s theory to pseudorepresentations.

We prove that if H is a finite-dimensional semisimple Hopf algebra acting on a PI-algebra R of characteristic 0, and R is either affine or algebraic over k, then the Jacobson radical of R is H-stable. Under the same hypotheses, we show that the smash product algebra R#H is semiprimitive provided that R is H-semiprime. More generally we show that the ‘finite’ Jacobson radical is H-stable, and that R#H is semiprimitive provided that R is H-semiprimitive and all irreducible representations of R are finite-dimensional. We also consider R#H when R is an FCR-algebra. Finally, we prove a general relationship between stability of the radical and semiprimeness of R#H; in particular if for a given H, any action of H stabilizes the Jacobson radical, then also any action of H stabilizes the prime radical.

The main result shows that if $R$ is a semiprime ring satisfying a polynomial identity, and if $Z(R)$ is the center of $R$, then card $R\,\le \,{{2}^{\text{card}\,Z\text{(}R\text{)}}}$. Examples show that this bound can be achieved, and that the inequality fails to hold for rings which are not semiprime.