Inverse limits of Banach algebras of polynomial growth, and $C^\infty$-functions of noncommuting variables

It is a long tradition in functional analysis to study Banach algebra elements (mostly operators) such that the norm of the function $s\mapsto exp(isb)$ grows like a polynomial in $|s|$ (here $b$ is the element and $s$ is a real number). However, Banach algebras of polynomial growth (i.e., those which entirely consist of such elements) were not considered so far. (Note that their theory is nontrivial only over the field of real numbers.) Banach algebras of polynomial growth satisfy some rather serious restrictions (e.g., the commutativity modulo the Jacobson radical and the nilpotency of the Jacobson radical). Nevertheless, the envelope functor that corresponds to this class of Banach algebras (which is similar to the Arens-Michael envelope) enables us to obtain Fréchet algebras whose elements can naturally be interpreted as $C^\infty$-functions of noncommuting variables.
In this talk, we pay special attention to the envelope of the universal enveloping algebra of a solvable Lie algebra. In particular, we discuss in detail the Heisenberg algebra and the two-dimensional nonabelian Lie algebra.