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

It is shown that a decomposition of a complex Lie group G G into a semidirect product generates a decomposition of the algebra of analytic functionals, A ( G ) \mathscr {A}(G) , into an analytic smash product in the sense of Pirkovskii. Also, sufficient conditions are found for a semidirect product to generate similar decompositions of certain Arens–Michael completions of A ( G ) \mathscr {A}(G) . The main result: if G 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 A ( G ) \mathscr {A}(G) into iterated analytic smash products. For the extreme cases, specifically, the envelope of A ( G ) \mathscr {A}(G) in the class of all Banach algebras (also known as the Arens–Michael envelope) and the envelope in the class of Banach PI-algebras (a new concept that is introduced in this article), decompositions are presented into iterated analytic smash products.