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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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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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