Completions of algebras of analytic functionals, PI algebras, and universal properties

In previous talks (September 2018 and May 2019) some properties of the algebra of analytic functionals on a connected complex Lie group and of its Arens-Michael envelope were discussed. Today other completions of this algebra (which are also Arens-Michael algebras) are of interest. I will explain how such completions can be constructed from a submultiplicative weight with use of a construction proposed by Akbarov. The main goal is to find those of them that satisfy universal properties related to the concept of weight exponentially distorted on a subgroup. We show that the exponential distortion can be achieved by using homomorphisms to Banach algebras that satisfy a polynomial identity (PI algebras). The proof is based both on purely algebraic results (the classical Kemer-Razmyslov theorem on the radical of a PI algebra and Bahturin's theorem on quotients of a universal enveloping algebra) and analytic (properties of generalized scalar elements and Turovskii's result on the inclusion of some commutant into the radical of a Banach algebra). The following result will be formulated: for each subgroup of a certain class there is a maximal weight exponentially distorted on it, and the corresponding completion satisfies some natural universal property. The main universal object that arises along the way is the envelope of a topological associative algebra in the class of Banach PI algebras (a weakened version of the Arens-Michael envelope). In particular, the `algebra of formally radical functions’ defined by Dosi is an example of such an envelope.