A real analogue of the Gel′fand-Neumark theorem
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- by Tamio Ono
- Proc. Amer. Math. Soc. 25 (1970), 159-160
- DOI: https://doi.org/10.1090/S0002-9939-1970-0257758-5
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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.References
- Horst Behncke, A note on the Gel′fand-Naĭmark conjecture, Comm. Pure Appl. Math. 23 (1970), 189–200. MR 257755, DOI 10.1002/cpa.3160230206
- Lars Ingelstam, Real Banach algebras, Ark. Mat. 5 (1964), 239–270 (1964). MR 172132, DOI 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
Bibliographic Information
- © Copyright 1970 American Mathematical Society
- 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