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Proceedings of the American Mathematical Society

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



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
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