Sheaves of noncommutative smooth and holomorphic functions associated to some solvable Lie algebras

In the earlier talk of 17.09.21 we introduced a class of real Banach algebras of polynomial growth and discussed envelopes assotiated to them. We also explicitly described the envelope of a universal enveloping algebra of a triangular Lie algebra (for abelian Lie algebras, the envelope is just the algebra of $C^\infty$ functions).
In the case where the Lie algebra is nilpotent, we define algebras corresponding to open subsets of the Gelfand spectrum. This yields a sheaf whose algebra of global sections is the above-mentioned envelope. However, if the Lie algebra is not nilpotent, the Gelfand spectrum is too small for defining algebras with similar properties. Nevertheless, we believe that such sheaves can be constructed in a more general setting. We consider a simple example of a situaton where this is indeed possible, namely the Lie algebra of the group of affine transformations of the line, and we construct a sheaf of algebras of polynomial growth on a special space of its representations (which is bigger than the Gelfand spectrum).
We also present a version of our constructions for noncommutative holomorphic functions. In particular, we obtain some generalizations of A. Dosiev's results for the holomorphic case.