Vestnik of Samara University. Natural Science SeriesVestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University644810.18287/2541-7525-2018-24-3-14-22UnknownON TOPOLOGICAL ALGEBRAS OF ANALYTIC FUNCTIONALS WITH A MULTIPLICATION DEFINED BY TRANSLATIONSIvanovaO. A.morenov@ssau.ruMelikhovS. N.morenov@ssau.ruSouthern Federal University1111201824314221101201911012019Copyright © 2018, Ivanova O.A., Melikhov S.N.2018<p>We define a multiplication convolution in the dual of a countable inductive limit <em>E </em>of weighted Frechet 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 <em>E</em>. 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 <em>E</em>. We show that this condition is essential for the validity of the last statement.</p>весовое пространство целых функций, алгебра аналитических функционалов, топологическая алгебра, коммутант, оператор свертки.weighted space of entire functions, algebra of analytic functionals, topological algebra, communant, convolution operator.[[1] Helemskii A.Ya. Gomologiia v banakhovykh i topologicheskikh algebrakh [Homology of Banach and topological algebras]. M.: Izd-vo MGU, 1986, 288 p. [in Russian].][[2] Helemskii A.Ya. Banakhovy i polinormirovannye algebry: obshchaia teoriia, predstavleniia, gomologii [Banach and multi-normed algebras: general theory, presentations, gomologies]. M.: Nauka, 1989, 466 p. 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