Decomposition of algebras of analytic functionals and of their completions

Algebras of analytic functionals on complex Lie groups are analogues of distribution algebras on real Lie groups. We consider the case of a connected linear group. Using the specifics of complex groups, a construction of Akbarov and the structure theory, we obtain a decomposition into an iterated analytic smash product in the sense of Pirkovskii, not only for the algebra of analytic functionals but also for its completions such as the Arens-Michael envelope and the envelope with respect to the class of Banach PI-algebras.