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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A real analogue of the Gel′fand-Neumark theorem
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by Tamio Ono
Proc. Amer. Math. Soc. 25 (1970), 159-160
DOI: https://doi.org/10.1090/S0002-9939-1970-0257758-5

Abstract:

Let $A$ be a real Banach $^{\ast }$-algebra enjoying the following three conditions: $||{x^{\ast }}x|| = ||{x^{\ast }}||\;||x||,\;Sp{x^{\ast }}x \geqq 0$, and $||{x^{\ast }}|| = ||x||\;(x \in A)$. It is shown, after Ingelstam, Palmer, and Behncke, as a real analogue of the Gelfand-Neumark theorem, that $A$ is isometrically $^{\ast }$-isomorphic onto a real ${C^{\ast }}$-algebra acting on a suitable real (or complex) Hilbert space. The converse is obvious.
References
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Bibliographic Information
  • © Copyright 1970 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 25 (1970), 159-160
  • MSC: Primary 46.65
  • DOI: https://doi.org/10.1090/S0002-9939-1970-0257758-5
  • MathSciNet review: 0257758