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Keywords:
$C(X)$-space; supra-additive; supra-multiplicative operator; realcompact space; realcompact
$C(X)$-space; supra-additive; supra-multiplicative operator; realcompact space; realcompact
Summary:
M. Radulescu proved the following result: Let $X$ be a compact Hausdorff topological space and ${\pi }\: C(X)\rightarrow C(X)$ a supra-additive and supra-multiplicative operator. Then ${\pi }$ is linear and multiplicative. We generalize this result to arbitrary topological spaces.
References:
[1] C. D. Aliprantis, O. Burkinshaw: Positive Operators. Academic Press, New York, 1985. MR 0809372
[2] Z. Ercan, S. Önal: A remark on the homomorphism on $C(X)$. Proc. Amer. Math. Soc. 133 (2005), 3609–3611. DOI 10.1090/S0002-9939-05-07930-X | MR 2163596
[3] K. P. Hart, J. Nagata, J. E. Vaughan: Encyclopedia of General Topology. Elsevier, Amsterdam, 2004. MR 2049453
[4] M. Radulescu: On a supra-additive and supra-multiplicative operator of $C(X)$. Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Sér. 24 (1980), 303–305. MR 0611909 | Zbl 0463.47034
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