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, A real ${B^{\ast }}$-algebra is ${C^{\ast }}$ iff it is hermitian, Notices Amer. Math. Soc. 16 (1969), 222-223. Abstract #663-468.
- 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
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Additional Information
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