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Proceedings of the American Mathematical Society
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
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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0746096-X
PII: S 0002-9939(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