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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Extending positive definite linear forms


Author: Jesús Gil de Lamadrid
Journal: Proc. Amer. Math. Soc. 91 (1984), 593-594
MSC: Primary 46Kxx
DOI: https://doi.org/10.1090/S0002-9939-1984-0746096-X
MathSciNet review: 746096
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Abstract: In the classical literature two properties, called here symmetry and bounded variation, of a positive definite linear form $ a'$ on an involutive Banach algebra $ \mathcal{A}$ are given. Together they form a necessary and sufficient condition that $ a'$ admit a positive definite linear extension to the involutive algebra obtained from $ \mathcal{A}$ by adjoining an identity. In this note we show that bounded variation alone suffices, in that it already implies symmetry.


References [Enhancements On Off] (What's this?)

  • [1] Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Company, Inc., Toronto-New York-London, 1953. MR 0054173
  • [2] M. A. Naĭmark, Normed rings, Reprinting of the revised English edition, Wolters-Noordhoff Publishing, Groningen, 1970. Translated from the first Russian edition by Leo F. Boron. MR 0355601

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0746096-X
Keywords: Involutive Banach algebras, positive definite forms, symmetry, bounded variation, Gelfand-Segal representation, cyclic vectors, extensibility
Article copyright: © Copyright 1984 American Mathematical Society