A real analogue of the Gelfand-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 be a real Banach -algebra enjoying the following three conditions: , and . It is shown, after Ingelstam, Palmer, and Behncke, as a real analogue of the Gelfand-Neumark theorem, that is isometrically -isomorphic onto a real -algebra acting on a suitable real (or complex) Hilbert space. The converse is obvious.

**[1]**H. Behncke,*A note on the Gel'fand-Naĭmark conjecture*, Comm. Math. Phys. (to appear). MR**0257755 (41:2404)****[2]**L. Ingelstam,*Real Banach algebras*, Ark. Mat.**5**(1964), 239-270. MR**33**#2358. MR**0172132 (30:2358)****[3]**T. Palmer,*A real*-*algebra is**iff it is hermitian*, Notices Amer. Math. Soc.**16**(1969), 222-223. Abstract #663-468.**[4]**T. Palmer,*Real*-*algebra*, Pacific J. Math. (to appear). MR**0270162 (42:5055)****[5]**C. Rickart,*General theory of Banach algebras*, The University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR**22**#5903. MR**0115101 (22:5903)**

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Additional Information

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

Keywords:
Real Banach -algebra,
isometrically -isomorphism,
real -algebra,
real Hilbert space,
complex Hilbert space,
complexification,
real -representation,
hermitian element,
involution,
hermitian involution,
symmetricity,
continuous involution,
sublinearity,
real linear functional,
skew adjointness,
real state,
-representation real Hilbert space,
-radical,
kernel,
bounded linear operator

Article copyright:
© Copyright 1970
American Mathematical Society