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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.

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  • [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 $ {B^{\ast}}$-algebra is $ {C^{\ast}}$ iff it is hermitian, Notices Amer. Math. Soc. 16 (1969), 222-223. Abstract #663-468.
  • [4] T. Palmer, Real $ {C^{\ast}}$-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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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
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