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

DOI:
https://doi.org/10.1090/S0002-9939-1970-0257758-5

MathSciNet review:
0257758

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Abstract | References | Similar Articles | Additional Information

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.

- Horst Behncke,
*A note on the Gel′fand-Naĭmark conjecture*, Comm. Pure Appl. Math.**23**(1970), 189–200. MR**257755**, DOI https://doi.org/10.1002/cpa.3160230206 - Lars Ingelstam,
*Real Banach algebras*, Ark. Mat.**5**(1964), 239–270 (1964). MR**172132**, DOI https://doi.org/10.1007/BF02591126
T. Palmer, - T. W. Palmer,
*Real $C^ *$-algebras*, Pacific J. Math.**35**(1970), 195–204. MR**270162** - Charles E. Rickart,
*General theory of Banach algebras*, The University Series in Higher Mathematics, D. van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0115101**

*A real*${B^{\ast }}$-

*algebra is*${C^{\ast }}$

*iff it is hermitian*, Notices Amer. Math. Soc.

**16**(1969), 222-223. Abstract #663-468.

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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