Ukrainian Mathematical Journal

, Volume 44, Issue 8, pp 1032–1035 | Cite as

Automatic continuity, bases, and radicals in metrizable algegbras

  • A. M. Plichko
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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.

Keywords

Banach Space Closed Subspace Orthogonal Basis Separable Banach Space Topological Algebra 

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Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • A. M. Plichko
    • 1
  1. 1.Institute of Applied Mechanics and MathematicsAcademy of Sciences of UkraineLviv

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