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A remark on the homomorphism on \(C(X)\). (English) Zbl 1087.46038

Summary: Let \(X\) be a real compact space. Without using the axiom of choice, we present a simple and direct proof that a non-zero homomorphism on \(C(X)\) is determined by a point.

MSC:

46J10 Banach algebras of continuous functions, function algebras
46E25 Rings and algebras of continuous, differentiable or analytic functions
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References:

[1] Richard M. Aron and Gerd H. Fricke, Homomorphisms on \?(\?), Amer. Math. Monthly 93 (1986), no. 7, 555. · Zbl 0651.46056
[2] Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. · Zbl 0684.54001
[3] Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. · Zbl 0093.30001
[4] Lyle E. Pursell, Comment: ”Homomorphisms on \?(\?)” [Amer. Math. Monthly 93 (1986), no. 7, 555; MR0856297 (87i:46109)] by R. M. Aron and G. H. Fricke, Amer. Math. Monthly 94 (1987), no. 7, 646.
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