Automatic continuity, bases, and radicals in metrizable algegbras
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.
English version (Springer): Ukrainian Mathematical Journal 44 (1992), no. 8, pp 1032-1035.
Citation Example: Plichko A. M. Automatic continuity, bases, and radicals in metrizable algegbras // Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1129–1132.