Automatic continuity, bases and radicals in the metrized algebras

  • A. M. Plichko Ін-т прикл. пробл. механіки і математики АН України, Львів

Abstract

The automatic continuity of a linear multiplicative operator T: X→Y, where X and Y are real complete metrizable algebras and Y semi-simple, is proved. It is shown that a complex Frechét algebra with absolute orthogonal basis (xi) (orthogonal in the sense that xixj=0 if i ≠ j) is a commutative symmetric involution algebra. Hence, we are able to derive the well-known result that every multiplicative linear functional defined on such an algebra is continuous. The concept of an orthogonal Markushevich basis in a topological algebra is introduced and is applied to show that, given an arbitrary closed subspace Y of a separable Banach space X, a commutative multiplicative operation whose radical is Y may be introduced on X. A theorem demonstrating the automatic continuity of positive functionals is proved.

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Published
08.09.1992
How to Cite
Plichko A. M. “Automatic Continuity, Bases and Radicals in the Metrized Algebras ”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 44, no. 8, Sept. 1992, pp. 1129-32, https://umj.imath.kiev.ua/index.php/umj/article/view/8046.
Section
Short communications