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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 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A real analogue of the Gel′fand-Neumark theorem

Author: Tamio Ono
Journal: Proc. Amer. Math. Soc. 25 (1970), 159-160
MSC: Primary 46.65
MathSciNet review: 0257758
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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.

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Keywords: Real Banach <IMG WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img9.gif" ALT="$^{\ast }$">-algebra, isometrically <IMG WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$^{\ast }$">-isomorphism, real <!– MATH ${C^{\ast }}$ –> <IMG WIDTH="31" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="${C^{\ast }}$">-algebra, real Hilbert space, complex Hilbert space, complexification, real <IMG WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img8.gif" ALT="$^{\ast }$">-representation, hermitian element, involution, hermitian involution, symmetricity, continuous involution, sublinearity, real linear functional, skew adjointness, real state, <IMG WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img7.gif" ALT="$^{\ast }$">-representation real Hilbert space, <IMG WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$^{\ast }$">-radical, kernel, bounded linear operator
Article copyright: © Copyright 1970 American Mathematical Society