Decomposition of the algebra of analytic functionals on a connected complex Lie group and its completions into iterated analytic smash products

We show that a decomposition of a complex Lie group $G$ into a semidirect product generates a decomposition of the algebra of analytic functionals, ${\mathscr A}(G)$, into an analytic smash product in the sense of Pirkovskii. Also we find sufficient conditions for a semidirect product to generate similar decompositions of certain Arens-Michael completions of ${\mathscr A}(G)$. The main result: if $G$ is connected, then its linearization admits a decomposition into an iterated semidirect product (with the composition series consisting of abelian factors and a semisimple factor) that induces a decomposition of algebras in a class of completions of ${\mathscr A}(G)$ into iterated analytic smash products. Considering the extreme cases, the envelope of ${\mathscr A}(G)$ in the class of all Banach algebras (aka the Arens-Michael envelope) and the envelope in the class Banach PI-algebras (a new concept that is introduced in this article), we decompose, in particular, these envelopes into iterated analytic smash products.