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A real analogue of the Gelfand-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: $ \vert\vert{x^{\ast}}x\vert\vert = \vert\vert{x^{\ast}}\vert\vert\;\vert\vert x\vert\vert,\;Sp{x^{\ast}}x \geqq 0$, and $ \vert\vert{x^{\ast}}\vert\vert = \vert\vert x\vert\vert\;(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.


References [Enhancements On Off] (What's this?)

  • [1] Horst Behncke, A note on the Gel′fand-Naĭmark conjecture, Comm. Pure Appl. Math. 23 (1970), 189–200. MR 0257755
  • [2] Lars Ingelstam, Real Banach algebras, Ark. Mat. 5 (1964), 239–270 (1964). MR 0172132
  • [3] 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.
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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0257758-5
Keywords: Real Banach $ ^{\ast}$-algebra, isometrically $ ^{\ast}$-isomorphism, real $ {C^{\ast}}$-algebra, real Hilbert space, complex Hilbert space, complexification, real $ ^{\ast}$-representation, hermitian element, involution, hermitian involution, symmetricity, continuous involution, sublinearity, real linear functional, skew adjointness, real state, $ ^{\ast}$-representation real Hilbert space, $ ^{\ast}$-radical, kernel, bounded linear operator
Article copyright: © Copyright 1970 American Mathematical Society