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Keywords:
locally m-convex algebra; $\Phi $-algebra; compact convergence topology
Summary:
Let $A$ be a uniformly closed and locally m-convex $\Phi $-algebra. We obtain internal conditions on $A$ stated in terms of its closed ideals for $A$ to be isomorphic and homeomorphic to $C_k(X)$, the $\Phi $-algebra of all the real continuous functions on a normal topological space $X$ endowed with the compact convergence topology.
References:
[1] F. W.  Anderson: Approximation in systems of real-valued continuous functions. Trans. Am. Math. Soc. 103 (1962), 249–271. DOI 10.1090/S0002-9947-1962-0136976-0 | MR 0136976 | Zbl 0175.14301
[2] R.  Bkouche: Couples spectraux et faisceaux associés. Applications aux anneaux de fonctions. Bull. Soc. Math. France 98 (1970), 253–295. MR 0477780 | Zbl 0201.37204
[3] W. A. Feldman, J. F.  Porter: The order topology for function lattices and realcompactness. Internat. J. Math. Math. Sci. 4 (1981), 289–304. DOI 10.1155/S0161171281000173 | MR 0613251
[4] I. M.  Gelfand: Normierte ringe. Rec. Math. Moscou, n. Ser. 9 (1941), 3–24. MR 0004726 | Zbl 0024.32002
[5] L.  Gillman, M.  Jerison: Rings of Continuous Functions. Grad. Texts in Math. Vol. 43. Springer-Verlag, New York, 1960. MR 0116199
[6] A. W. Hager: On inverse-closed subalgebras of  $C(X)$. Proc. London Math. Soc. 19 (1969), 233–257. MR 0244948 | Zbl 0169.54005
[7] M. Henriksen: Unsolved problems on algebraic aspects of  $C(X)$. In: Rings of Continuous Functions. Lecture Notes in Pure and Appl. Math. Vol. 95, M.  Dekker, New York, 1985, pp. 195–202. MR 0789271 | Zbl 0587.54028
[8] M. Henriksen, D. J.  Johnson: On the structure of a class of Archimedean lattice-ordered algebras. Fundam. Math. 50 (1961), 73–94. MR 0133698
[9] C. B.  Huijsmans, B. de Pagter: Ideal theory in $f$-algebras. Trans. Am. Math. Soc. 269 (1982), 225–245. MR 0637036
[10] D. J. Johnson: A structure theory for a class of lattice-ordered rings. Acta Math. 104 (1960), 163–215. DOI 10.1007/BF02546389 | MR 0125141 | Zbl 0094.25305
[11] F.  Montalvo, A.  Pulgarín, B. Requejo: Order topologies on $l$-algebras. Topology Appl. 137 (2004), 225–236. DOI 10.1016/S0166-8641(03)00212-8 | MR 2057889
[12] P. D. Morris, D. E.  Wulbert: Functional representation of topological algebras. Pac. J. Math. 22 (1967), 323–337. MR 0213876
[13] J.  Muñoz, J. M.  Ortega: Sobre las álgebras localmente convexas. Collect. Math. 20 (1969), 127–149.
[14] D.  Plank: Closed $l$-ideals in a class of lattice-ordered algebras. Ill. J. Math. 15 (1971), 515–524. MR 0280423
[15] A.  Pulgarín: A characterization of  $C_k(X)$ as a Fréchet $f$-algebra. Acta Math. Hung. 88 (2000), 133–138. DOI 10.1023/A:1006764913661 | Zbl 0988.54018
[16] B. Requejo: A characterization of the topology of compact convergence on  $C(X)$. Topology Appl. 77 (1997), 213–219. DOI 10.1016/S0166-8641(96)00143-5 | MR 1451653
[17] B. Requejo: Localización Topológica. Publ. Dpto. Mat. Unex, Vol. 32, Badajoz, 1995. MR 1387595
[18] H. H.  Schaefer: Topological Vector Spaces. Grad. Text in Math. Vol. 3. Springer-Verlag, New York, 1971. MR 0342978
[19] H.  Tietze: Über Funktionen, die auf einer abgeschlossenen Menge stetig sind. J. für Math. 145 (1914), 9–14.
[20] S.  Warner: The topology of compact convergence on continuous function spaces. Duke Math. J. 25 (1958), 265–282. MR 0102735 | Zbl 0081.32802
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