On topological algebras of analytic functionals with a multiplication defined by translations

We define a multiplication — convolution in the dual of a countable inductive limit $E$ of weighted Fréchet spaces of entire functions of several variables. This algebra is isomorphic to the commutant of the system of partial derivatives in the algebra of all continuous linear operators in $E$. In the constructed algebra of analytic functionals in two pure cases a topology is defined. With this topology the mentioned algebra is topological and it is now topologically isomorphic to the considered commutant with its natural operator topology. It is proved that in this pure situations the present algebra has no zero divisors provided that polynomials are dense in $E$. We show that this condition is essential for the validity of the last statement.